Problem of the Month (July 2019)

This month we consider several problems involving products, quotients, and divisors.

Problem #1:

There are no numbers of more than one digit that are the product of their own digits. But if we are willing to multiply by one more digit, there are some. And if we are willing to repeat the process, multiplying by possibly different numbers in order to obtain the original number, there are even more. Let 1 ≤ ai ≤ 9 be a finite sequence of positive integers. Start with a positive integer n. In step i, replace the current number with the product of its digits times ai. Which starting numbers n can result in a sequence ending in n? For example, the number 28 can return to itself with multipliers {4,3,2}: 2×8×4 = 64, 6×4×3=72, 7×2×2=28. For a given n, what is the shortest and smallest sequence of ai which accomplish this?

Looked at another way, 9 is the largest number that can return using ai ≤ 1, 36 is the largest using ai ≤ 2, and 288 is the largest using ai ≤ 3. What is the largest number that has a sequence ai ≤ n?

What numbers can return using the same ai each time?

Problem #2:

Some numbers n have the property that if some occurrences of a particular digit string are replaced with the same digit, the resulting number is a factor of n. For example, for the substitution 30 → 7, we have the solutions 304130825 / 7417825 = 41, 430830 / 4787 = 90, 6303075 / 67775 = 93, and 1130304 / 11774 = 96. What are the small solutions? Which pairs can you prove have no solutions? What patterns are there?

Problem #3:

Given numbers n>10 and k>1, it is not hard to find a string of digits so that the number formed by every k consecutive digits is divisible by n. But only some pairs n and k have a unique longest sequence. For example, for n=17 and k=2, the only sequence of length 5 is 68517, since 68, 85, 51, and 17 are all multiples of 17. What are the longest unique solutions for small n and k?

Problem #4:

Let d(n) denote the number of divisors of n. What is the smallest solution to x+y=z if we are given the values of d(x), d(y), and d(z)?

ANSWERS

Problem #1:

Here are the numbers that can return under iteration with all ai ≤ 9. Joe DeVincentis confirmed there are no others.

One Step
nMultipliers
1{1}
2{1}
3{1}
4{1}
5{1}
6{1}
7{1}
8{1}
9{1}
12{6}
15{3}
24{3}
36{2}
128{8}
135{9}
144{9}
175{5}
384{4}
672{8}
735{7}
1575{9}

Two Steps
nMultipliers
14{3,7}
16{3,2}
18{2,3}
32{3,4}
35{1,7}
42{2,7}
48{1,8}
64{1,8}
72{3,9}
75{1,5}
96{3,8}
98{1,7}
126{3,7}
162{3,9}
168{9,7}
216{8,4}
224{3,7}
288{3,3}
315{5,9}
324{4,6}
336{7,2}
378{2,7}
432{4,8}
768{8,1}
864{2,9}
896{4,8}
1176{9,7}
1728{8,4}
2688{1,8}
69984{9,6}
139968{6,9}
Three Steps
nMultipliers
28{4,3,2}
84{2,3,6}
112{2,7,7}
192{1,6,6}
448{7,4,4}
486{2,1,9}
784{4,4,7}
1344{2,7,8}
6144{8,8,8}
Four Steps
nMultipliers
648{1,4,7,9}
882{7,3,9,7}
972{7,7,3,9}
1296{6,9,8,3}
3888{4,3,7,9}
6912{6,4,8,9}
13824{9,8,9,9}
Five Steps
nMultipliers
294{2,6,7,7,7}
1792{7,7,4,7,8}
9216{6,9,8,9,6}
12288{7,7,7,9,8}
Six Steps
nMultipliers
1134{4,9,7,3,9,9}

Here are the largest known numbers that can return using ai ≤ n:

nNumberMultipliersAuthor
19{1}
264{2,2}(Marc Lapierre)
3384{3,3}(Marc Lapierre)
4384{4}
5384{4}
61344{6,6,4}(Marc Lapierre)
71728{4,7,4}(Marc Lapierre)
86144{8,8,8}
9139968{6,9}
11??
12??
13??

Here are the known numbers that can return using the same ai each time:

MultipliersCycles
{1}1, 2, 3, 4, 5, 6, 7, 8, 9
{2}36
{2,2}48 → 64
{2,2,2,2}4 → 8 → 16 → 12
{3}15, 24
{3,3}288 → 384
{4}384
{4,4}24 → 32
{4,4,4,4,4}48 → 128 → 64 → 96 → 216
{5}175
{6}12
{6,6,6}96 → 324 → 144
{7}735
{7,7,7}14 → 28 → 112
{8}128, 672
{8,8,8}768 → 2688 → 6144
{9}135, 144, 1575
{11}11
{11,11,11,11,11}66 → 396 → 1782 → 1232 → 132
{12}1296, 139968
{12,12,12}144 → 192 → 216
{13}624, 3276, 1886976
{13,13}91 → 117
{14}224
{17}816
{18}216, 432, 34992
{19}1197, 12768
{19,19}133 → 171
{21}315
{22}132, 3168
MultipliersCycles
{23}115, 6624, 8832
{23,23,23}368 → 3312 → 414
{27}2916
{28}1176, 1344
{29}3915, 739935
{31}93744
{31,31}63488 → 142848
{33}51975
{33,33}7392 → 12474
{34}78962688
{36}82944
{37}1184
{38,38}1824 → 2432
{41}31488
{44}77616
{44,44}2816 → 4224
{44,44,44,44}176 → 1848 → 11264 → 2112
{45}77175
{46}4416, 973728
{48}12288
{49}1715, 18816
{49,49,49}392 → 2646 → 14112
{51}612, 182784
{52}312
{52,52,52}936 → 8424 → 13312
{53}212
{53,53}186984 → 732672
{56}112
{57}344736, 1378944
{59}1416, 3746736
MultipliersCycles
{61}1264896, 26984448
{61,61}184464 → 187392
{61,61,61,61}7686 → 122976 → 92232 → 13176
{63}3483648, 4644864
{64}6144, 6912, 9289728
{66}4224
{67}2144, 35175
{68,68,68,68}7344 → 22848 → 69632 → 132192
{69}9315, 13248
{71,71}27832 → 47712
{72}13824
{74}1332, 28416
{76}1329696
{77}16632, 33264
{79}4368384
{81}11664, 23328, 186624, 248832, 2239488
{82}1983744
{83}167328, 382464
{84}163296
{86}13832
{86,86}24768 → 231168
{89}9612, 89712, 179424
{91}91728
{93}2232, 437472, 1687392, 3796632, 47246976
{94}48128, 144384
{94,94}1989792 → 7674912
{96}18432, 442368
{97}21728
{98}42336, 22478848, 28449792
{98,98,98}84672 → 263424 → 112896

Problem #2:

Here are the known solutions where 1 or 2 digit strings yield quotients that have no more than 2 digits. I only list the smallest pair to give any given quotient, as 0's can always be added to the end of both numbers. Thanks to Joe DeVincentis for finding several solutions I missed, and for sending me HTML code.

0123456789
1










2
2 / 1 = 2 









3
3 / 1 = 3 









4
45 / 15 = 3
4 / 1 = 4 
		4 / 2 = 2 








5
5 / 1 = 5 









6
625 / 125 = 5
6 / 1 = 6 
		6 / 2 = 3 
6 / 3 = 2 







7
75 / 15 = 5
72 / 12 = 6
7 / 1 = 7 
		75 / 25 = 3 








8
875 / 175 = 5
84 / 14 = 6
8 / 1 = 8 
		8 / 2 = 4 

8 / 4 = 2 






9
96 / 16 = 6
9 / 1 = 9 
		
9 / 3 = 3 







00100344827586200625 / 1034482758620625 = 97 (JD)
100125 / 11125 = 9 (JD)
20025 / 2225 = 9 (JD)
300375 / 33375 = 9 (JD)
4005 / 445 = 9 (JD)
200 / 25 = 8 (JD)
500625 / 55625 = 9 (JD)
		1008 / 168 = 6 (JD)
60075 / 6675 = 9 (JD)
200700 / 2676 = 75 (JD)
		700875 / 77875 = 9 (JD)

940054005 / 9495495 = 99 (JD)

011010101 / 10001 = 101 (JD)
10125 / 1125 = 9 (JD)
101101 / 1111 = 91 (JD)
		201375 / 22375 = 9 (JD)
3015 / 335 = 9 (JD)
101101 / 1313 = 77 (JD)
		1015 / 145 = 7 (JD)
401625 / 44625 = 9 (JD)
		50175 / 5575 = 9 (JD)
601875 / 66875 = 9 (JD)
3011013 / 37173 = 81 (JD)
8501764044015 / 85876404485 = 99 (JD)
801 / 89 = 9 (JD)
970101 / 9799 = 99 (JD)

022020202 / 20002 = 101 (JD)
102375 / 11375 = 9 (JD)
2025 / 225 = 9 (JD)
202202 / 2222 = 91 (JD)
		302625 / 33625 = 9 (JD)
40275 / 4475 = 9 (JD)
502875 / 55875 = 9 (JD)
202202 / 2626 = 77 (JD)
102 / 17 = 6 (JD)
702 / 78 = 9 (JD)
8602902 / 86898 = 99 (JD)
		802125 / 89125 = 9 (JD)

033030303 / 30003 = 101 (JD)
1035 / 115 = 9 (JD)
203625 / 22625 = 9 (JD)
30375 / 3375 = 9 (JD)
1036032 / 13632 = 76 (JD)
303303 / 3333 = 91 (JD)
		403875 / 44875 = 9 (JD)
203070422535211203975 / 2507042253521125975 = 81 (JD)
1903200321875 / 19620621875 = 97 (JD)
1032 / 172 = 6 (JD)
603 / 67 = 9 (JD)
7503903 / 75797 = 99 (JD)
		703125 / 78125 = 9 (JD)
203 / 29 = 7 (JD)
80325 / 8925 = 9 (JD)
303303 / 3939 = 77 (JD)
310303 / 3199 = 97 (JD)

044040404 / 40004 = 101 (JD)
104625 / 11625 = 9 (JD)
20475 / 2275 = 9 (JD)
104 / 13 = 8 (JD)
304875 / 33875 = 9 (JD)
		404404 / 4444 = 91 (JD)
2042112049875 / 25211259875 = 81 (JD)
504 / 56 = 9 (JD)
6404904 / 64696 = 99 (JD)
		1044 / 174 = 6 (JD)
604125 / 67125 = 9 (JD)
		304 / 38 = 8 (JD)
70425 / 7825 = 9 (JD)
		804375 / 89375 = 9 (JD)

055050505 / 50005 = 101 (JD)
10575 / 1175 = 9 (JD)
205875 / 22875 = 9 (JD)
60502325581056 / 630232558136 = 96 (JD)
3905055 / 39445 = 99 (JD)
105 / 15 = 7 (JD)
405 / 45 = 9 (JD)
505505 / 5555 = 91 (JD)
5305905 / 53595 = 99 (JD)
		505125 / 56125 = 9 (JD)
1056 / 176 = 6 (JD)
60525 / 6725 = 9 (JD)
		705375 / 78375 = 9 (JD)
2005056 / 20886 = 96 (JD)
		8055 / 895 = 9 (JD)

066060606 / 60006 = 101 (JD)
106875 / 11875 = 9 (JD)
10636842105064 / 123684210524 = 86 (JD)
806100615625 / 8310315625 = 97 (JD)
306 / 34 = 9 (JD)
4206906 / 42494 = 99 (JD)
		406125 / 45125 = 9 (JD)
50625 / 5625 = 9 (JD)
606606 / 6666 = 91 (JD)
		1068 / 178 = 6 (JD)
606375 / 67375 = 9 (JD)
		7065 / 785 = 9 (JD)
2065 / 295 = 7 (JD)
806625 / 89625 = 9 (JD)

077070707 / 70007 = 101 (JD)
207379307 / 2137931 = 97 (JD)
1807584075 / 18258425 = 99 (JD)
207 / 23 = 9 (JD)
10720755 / 132355 = 81 (JD)
3107907 / 31393 = 99 (JD)
		307125 / 34125 = 9 (JD)
40725 / 4525 = 9 (JD)
2070765 / 25565 = 81 (JD)
		507375 / 56375 = 9 (JD)
6075 / 675 = 9 (JD)
707707 / 7777 = 91 (JD)
		707625 / 78625 = 9 (JD)
80775 / 8975 = 9 (JD)

088080808 / 80008 = 101 (JD)
608460674085 / 6146067415 = 99 (JD)
108 / 12 = 9 (JD)
30809088 / 320928 = 96 (JD)
2008908 / 20292 = 99 (JD)
		208125 / 23125 = 9 (JD)
30825 / 3425 = 9 (JD)
1085 / 155 = 7 (JD)
408375 / 45375 = 9 (JD)
		208 / 26 = 8 (JD)
5085 / 565 = 9 (JD)
		608625 / 67625 = 9 (JD)
108 / 18 = 6 (JD)
70875 / 7875 = 9 (JD)
808808 / 8888 = 91 (JD)
26080875 / 268875 = 97 (JD)
		808875 / 89875 = 9 (JD)

099090909 / 90009 = 101 (JD)
909909 / 9191 = 99 (JD)
109125 / 12125 = 9 (JD)
20925 / 2325 = 9 (JD)
309375 / 34375 = 9 (JD)
4095 / 455 = 9 (JD)
509625 / 56625 = 9 (JD)
10960092 / 166062 = 66 (JD)
		60975 / 6775 = 9 (JD)
1092 / 182 = 6 (JD)
709875 / 78875 = 9 (JD)
		909909 / 9999 = 91 (JD)

10110 / 10 = 11
1010101 / 10001 = 101 (JD)
		108 / 18 = 6
105 / 15 = 7
10125 / 1125 = 9
10 / 1 = 10
101101 / 1111 = 91 
		10 / 2 = 5
11025 / 1225 = 9 (JD)
3103104 / 32324 = 96 (JD)
		105 / 35 = 3 (JD)
210375 / 23375 = 9 (JD)
		3105 / 345 = 9 (JD)
1101088 / 14488 = 76 
		10 / 5 = 2
410625 / 45625 = 9 (JD)
23107910 / 235795 = 98
101010 / 555 = 182 (JD)
10710810 / 57585 = 186 (JD)
		51075 / 5675 = 9 (JD)
610875 / 67875 = 9 (JD)
101075 / 7775 = 13
101010 / 777 = 130 (JD)
		1104 / 184 = 6 (JD)
101101 / 9191 = 11 

111111 / 101 = 11
11111111 / 110011 = 101 (JD)
111111 / 1001 = 111 (JD)
		1125 / 125 = 9
11 / 1 = 11
1111 / 11 = 101 (JD)
		1125 / 225 = 5
111375 / 12375 = 9 (JD)
119011725 / 2902725 = 41
11116875 / 226875 = 49 (JD)
111111 / 1221 = 91 
		2115 / 235 = 9 (JD)
11116875 / 336875 = 33 (JD)
		311625 / 34625 = 9 (JD)
11117115 / 411745 = 27 
		41175 / 4575 = 9 (JD)
511875 / 56875 = 9 (JD)
111111 / 777 = 143 (JD)
1116 / 186 = 6 (JD)
711 / 79 = 9 (JD)
71111 / 799 = 89 

121122 / 102 = 11
12121212 / 120012 = 101 (JD)
		12375 / 1375 = 9
121 / 11 = 11
12 / 1 = 12
1121201 / 11101 = 101 (JD)
12089121 / 108911 = 111 (JD)
		125 / 25 = 5
12 / 2 = 6
1125 / 125 = 9 (JD)
121125 / 2125 = 57 
		12 / 3 = 4
212625 / 23625 = 9 (JD)
212121 / 2331 = 91 
		12 / 4 = 3
112 / 14 = 8 (JD)
31275 / 3475 = 9 (JD)
		412875 / 45875 = 9 (JD)
12 / 6 = 2
112 / 16 = 7 (JD)
1312128 / 13668 = 96
18128912 / 186896 = 97
121212 / 666 = 182 (JD)
		121212 / 777 = 156 (JD)
1128 / 188 = 6 (JD)
612 / 68 = 9 (JD)
751212 / 7588 = 99 
		312 / 39 = 8 (JD)
712125 / 79125 = 9 (JD)

131133 / 103 = 11
13131313 / 130013 = 101 (JD)
		135 / 15 = 9
132 / 12 = 11
13 / 1 = 13
1381376 / 18176 = 76
13690134 / 169014 = 81
1331382 / 13182 = 101 (JD)
13188132 / 118812 = 111 (JD)
1313625 / 11625 = 113 (JD)
1308131 / 10811 = 121 (JD)
		1375 / 275 = 5
132 / 22 = 6
113625 / 12625 = 9 (JD)
13681362 / 268262 = 51 
		21375 / 2375 = 9 (JD)
135 / 45 = 3
313875 / 34875 = 9 (JD)
1328133 / 42843 = 31
313131 / 3441 = 91 
		
3134473132 / 36447362 = 86 (JD)
513 / 57 = 9 (JD)
131325 / 7725 = 17
131313 / 777 = 169 (JD)
		613125 / 68125 = 9 (JD)
71325 / 7925 = 9 (JD)

141144 / 104 = 11
14141414 / 140014 = 101 (JD)
		14625 / 1625 = 9
143 / 13 = 11
14 / 1 = 14
3141403 / 31103 = 101 (JD)
14287143 / 128713 = 111 (JD)
14145 / 115 = 123 (JD)
1414224 / 11224 = 126 (JD)
		144 / 24 = 6
14 / 2 = 7
11475 / 1275 = 9 (JD)
1491444 / 29244 = 51
1435147 / 23527 = 61 
		214875 / 23875 = 9 (JD)
1454814275 / 35483275 = 41 
		1140814125 / 14084125 = 81 (JD)
414141 / 4551 = 91 
414 / 46 = 9 (JD)
14 / 7 = 2
514125 / 57125 = 9 (JD)
141414 / 777 = 182 (JD)
		61425 / 6825 = 9 (JD)
14144 / 884 = 16 
		114 / 19 = 6 (JD)
714375 / 79375 = 9 (JD)

151155 / 105 = 11
15151515 / 150015 = 101 (JD)
		1575 / 175 = 9
154 / 14 = 11
15 / 1 = 15
1597158 / 19718 = 81
4151504 / 41104 = 101 (JD)
15386154 / 138614 = 111 (JD)
151575 / 1175 = 129 (JD)
		156 / 26 = 6
115875 / 12875 = 9 (JD)
151159 / 2129 = 71 
		15 / 3 = 5
315 / 35 = 9 (JD)
		11522158875 / 142248875 = 81 (JD)
15 / 5 = 3
315 / 35 = 9 (JD)
361516815 / 3651685 = 99 
		1155 / 165 = 7 (JD)
415125 / 46125 = 9 (JD)
515151 / 5661 = 91 
		51525 / 5725 = 9 (JD)
1526154 / 72674 = 21
111541115 / 1174117 = 95
151515 / 777 = 195 (JD)
		615375 / 68375 = 9 (JD)
1152 / 192 = 6 (JD)
7155 / 795 = 9 (JD)
271515125 / 2799125 = 97 

161166 / 106 = 11
16161616 / 160016 = 101 (JD)
		16875 / 1875 = 9
165 / 15 = 11
1625 / 125 = 13
16 / 1 = 16
5161605 / 51105 = 101 (JD)
16485165 / 148515 = 111 (JD)
1641516 / 14151 = 116 (JD)
		168 / 28 = 6
16 / 2 = 8 
		1625 / 325 = 5
1716165 / 17335 = 99 
		16 / 4 = 4
216 / 24 = 9 (JD)
916165 / 9445 = 97 
		165 / 55 = 3
316125 / 35125 = 9 (JD)
161675 / 5575 = 29 
		41625 / 4625 = 9 (JD)
1624168 / 62468 = 26
21616875 / 266875 = 81 
		216 / 27 = 8 (JD)
516375 / 57375 = 9 (JD)
4163016 / 47307 = 88 (JD)
616161 / 6771 = 91 
		16 / 8 = 2
6165 / 685 = 9 (JD)
161616 / 888 = 182 (JD)
		1164 / 194 = 6 (JD)
716625 / 79625 = 9 (JD)

171177 / 107 = 11
17171717 / 170017 = 101 (JD)
		176 / 16 = 11
17 / 1 = 17
6171706 / 61106 = 101 (JD)
17584176 / 158416 = 111 (JD)
1722174 / 12214 = 141 (JD)
		175 / 25 = 7
17176 / 226 = 76 
		175 / 35 = 5
117 / 13 = 9 (JD)
171768 / 3368 = 51
17174817 / 173483 = 99 
		217125 / 24125 = 9 (JD)
1719317575 / 41934575 = 41 
		31725 / 3525 = 9 (JD)
17174 / 554 = 31
217175 / 2555 = 85 
		417375 / 46375 = 9 (JD)
178741784 / 6874684 = 26 
		5175 / 575 = 9 (JD)
617625 / 68625 = 9 (JD)
717171 / 7881 = 91 
		1176 / 196 = 6 (JD)
71775 / 7975 = 9 (JD)

181188 / 108 = 11
18181818 / 180018 = 101 (JD)
		187 / 17 = 11
180625 / 10625 = 17 (JD)
18 / 1 = 18
7181807 / 71107 = 101 (JD)
18683187 / 168317 = 111 (JD)
		18 / 2 = 9
1862188 / 26228 = 71
181845 / 2245 = 81 
		1875 / 375 = 5
18 / 3 = 6
118125 / 13125 = 9 (JD)
		21825 / 2425 = 9 (JD)
181835 / 4435 = 41 
		318375 / 35375 = 9 (JD)
18 / 6 = 3
4185 / 465 = 9 (JD)
		518625 / 57625 = 9 (JD)
61875 / 6875 = 9 (JD)
18 / 9 = 2
1188 / 198 = 6 (JD)
718875 / 79875 = 9 (JD)
11831865 / 193965 = 61 (JD)
818181 / 8991 = 91
181818 / 999 = 182 (JD)

191199 / 109 = 11
19191919 / 190019 = 101 (JD)
		198 / 18 = 11
195 / 15 = 13
192 / 12 = 16
19125 / 1125 = 17
19 / 1 = 19
8191908 / 81108 = 101 (JD)
19782198 / 178218 = 111 (JD)
19698192 / 169812 = 116 (JD)
19621965 / 162165 = 121 (JD)
1941975 / 14175 = 137 (JD)
1919925 / 11925 = 161 (JD)
		19125 / 2125 = 9 
192 / 32 = 6
11925 / 1325 = 9 (JD)
199021992 / 3902392 = 51 
		219375 / 24375 = 9 (JD)
3195 / 355 = 9 (JD)
195 / 65 = 3
419625 / 46625 = 9 (JD)
		119 / 17 = 7 (JD)
51975 / 5775 = 9 (JD)
		619875 / 68875 = 9 (JD)
319436619718219875 / 3943661971829875 = 81 (JD)

20220 / 20 = 11
120 / 10 = 12 (JD)
2020202 / 20002 = 101 (JD)
		209 / 19 = 11
201875 / 11875 = 17
20 / 1 = 20
9202009 / 91109 = 101 (JD)
20881209 / 188119 = 111 (JD)
202048 / 1148 = 176 (JD)
		2025 / 225 = 9
20 / 2 = 10
2072064 / 27264 = 76
20535201 / 253521 = 81
202202 / 2222 = 91 
		204 / 34 = 6
120375 / 13375 = 9 (JD)
		20 / 4 = 5
2205 / 245 = 9 (JD)
6206208 / 64648 = 96 (JD)
		20 / 5 = 4
120 / 15 = 8 (JD)
320625 / 35625 = 9 (JD)
		42075 / 4675 = 9 (JD)
520875 / 57875 = 9 (JD)
202072 / 7772 = 26 
		8201205 / 88185 = 93 (JD)
222020875 / 2288875 = 97 
		182065116278208 / 1896511627898 = 96 (JD)

212211 / 201 = 11
12121175 / 100175 = 121 (JD)
		121 / 11 = 11 (JD)
2125 / 125 = 17
21 / 1 = 21
921221 / 9121 = 101 (JD)
		21375 / 2375 = 9 
216 / 36 = 6
21 / 3 = 7
1215 / 135 = 9 (JD)
21529218 / 352938 = 61
121212 / 1332 = 91
5217921 / 53793 = 97 
		2125 / 425 = 5
221625 / 24625 = 9 (JD)
2112114 / 41414 = 51 
		32175 / 3575 = 9 (JD)
2112115 / 51515 = 41 
		421875 / 46875 = 9 (JD)
62182125 / 668625 = 93 (JD)
		21 / 7 = 3 
2102152 / 80852 = 26
1213212 / 18382 = 66 (JD)
		621 / 69 = 9 (JD)

222222 / 202 = 11
222222 / 2002 = 111 (JD)
		1221 / 111 = 11 (JD)
2275 / 175 = 13
225 / 15 = 15
224 / 14 = 16
223125 / 13125 = 17
2205 / 105 = 21 (JD)
22 / 1 = 22
11222211 / 111111 = 101 (JD)
22981224 / 198114 = 116 (JD)
22822128 / 181128 = 126 (JD)
225922 / 1591 = 142 (JD)
222222 / 1221 = 182 (JD)
		225 / 25 = 9
22 / 2 = 11
2222 / 22 = 101 (JD)
		228 / 38 = 6
122625 / 13625 = 9 (JD)
		225 / 45 = 5
22275 / 2475 = 9 (JD)
22612275 / 461475 = 49
222222 / 2442 = 91 
		322875 / 35875 = 9 (JD)
2295652173913043422125 / 69565217391304346125 = 33 (JD)
225 / 75 = 3
1225 / 175 = 7 (JD)
		522 / 58 = 9 (JD)
622125 / 69125 = 9 (JD)
62222125 / 699125 = 89 

232233 / 203 = 11 
1232 / 112 = 11 (JD)
23375 / 1375 = 17
231 / 11 = 21
23 / 1 = 23
12232312 / 121112 = 101 (JD)
		23625 / 2625 = 9
231 / 21 = 11
1232301 / 12201 = 101 (JD)
2323125 / 22125 = 105 (JD)
		12375 / 1375 = 9 (JD)
2375 / 475 = 5
223875 / 24875 = 9 (JD)
		23802345 / 580545 = 41
323232 / 3552 = 91 
		44238539325842235 / 446853932584265 = 99 (JD)
423 / 47 = 9 (JD)
532323 / 5377 = 99 
		523125 / 58125 = 9 (JD)
62325 / 6925 = 9 (JD)
3232305 / 39905 = 81 (JD)
1823232 / 18992 = 96 

242244 / 204 = 11 
1243 / 113 = 11 (JD)
244375 / 14375 = 17
2415 / 115 = 21
24 / 1 = 24
13242413 / 131113 = 101 (JD)
		2475 / 275 = 9
242 / 22 = 11
24 / 2 = 12
2242402 / 22202 = 101 (JD)
		245 / 35 = 7
24 / 3 = 8
124875 / 13875 = 9 (JD)
		24 / 4 = 6 
29243932245 / 295393255 = 99 (JD)
24 / 6 = 4
324 / 36 = 9 (JD)
424242 / 4662 = 91 
		424125 / 47125 = 9 (JD)
24 / 8 = 3
224 / 28 = 8 (JD)
52425 / 5825 = 9 (JD)
312424 / 3188 = 98 
		624375 / 69375 = 9 (JD)

252255 / 205 = 11 
1254 / 114 = 11 (JD)
256 / 16 = 16
255 / 15 = 17
252 / 12 = 21
25 / 1 = 25
14252514 / 141114 = 101 (JD)
		25875 / 2875 = 9
253 / 23 = 11
2552573 / 25273 = 101 (JD)
		225488256 / 2348836 = 96 (JD)
252 / 42 = 6 
25 / 5 = 5
225 / 25 = 9 (JD)
		325125 / 36125 = 9 (JD)
525252 / 5772 = 91 
		42525 / 4725 = 9 (JD)
525252 / 5772 = 91 (JD)
		255 / 85 = 3
525375 / 58375 = 9 (JD)
		6255 / 695 = 9 (JD)

262266 / 206 = 11
12626208 / 100208 = 126 (JD)
		1265 / 115 = 11 (JD)
265625 / 15625 = 17
2625 / 125 = 21
26 / 1 = 26
15262615 / 151115 = 101 (JD)
		264 / 24 = 11
26 / 2 = 13
4262604 / 42204 = 101 (JD)
2616262 / 21622 = 121 (JD)
		262267875 / 3237875 = 81 (JD)
1261395264 / 13139534 = 96 (JD)
		264 / 44 = 6
126 / 14 = 9 (JD)
		2625 / 525 = 5
226125 / 25125 = 9 (JD)
		32625 / 3625 = 9 (JD)
426375 / 47375 = 9 (JD)
126 / 18 = 7 (JD)
5265 / 585 = 9 (JD)
2656266 / 85686 = 31
626262 / 6882 = 91
182627526 / 1882758 = 97 
		626625 / 69625 = 9 (JD)

272277 / 207 = 11
3272725 / 30025 = 109 (JD)
		1276 / 116 = 11 (JD)
27625 / 1625 = 17
273 / 13 = 21
2704 / 104 = 26 (JD)
27 / 1 = 27
16272716 / 161116 = 101 (JD)
		275 / 25 = 11
5272705 / 52205 = 101 (JD)
		27 / 3 = 9
2762752 / 36352 = 76
2727675 / 33675 = 81 
		276 / 46 = 6
127125 / 14125 = 9 (JD)
327275 / 3445 = 95 
		275 / 55 = 5
22725 / 2525 = 9 (JD)
27362724 / 536524 = 51 
		327375 / 36375 = 9 (JD)
2727525 / 66525 = 41 
		4275 / 475 = 9 (JD)
527625 / 58625 = 9 (JD)
27 / 9 = 3
62775 / 6975 = 9 (JD)
2272704 / 29904 = 76 (JD)
727272 / 7992 = 91
23272725 / 239925 = 97 

282288 / 208 = 11 
1287 / 117 = 11 (JD)
288 / 18 = 16
286875 / 16875 = 17
285 / 15 = 19
2835 / 135 = 21
28125 / 1125 = 25
2808 / 108 = 26 (JD)
28 / 1 = 28
17282817 / 171117 = 101 (JD)
2892858 / 19158 = 151 (JD)
28728 / 171 = 168 (JD)
		286 / 26 = 11
28 / 2 = 14
6282806 / 62206 = 101 (JD)
2828448 / 22448 = 126 (JD)
		28125 / 3125 = 9 
288 / 48 = 6
28 / 4 = 7
12825 / 1425 = 9 (JD)
52828625 / 544625 = 97 
		2875 / 575 = 5
228375 / 25375 = 9 (JD)
282846 / 5546 = 51 
		128 / 16 = 8 (JD)
3285 / 365 = 9 (JD)
1728328 / 17636 = 98 
		28 / 7 = 4
428625 / 47625 = 9 (JD)
		52875 / 5875 = 9 (JD)
285 / 95 = 3
628875 / 69875 = 9 (JD)

292299 / 209 = 11 
1298 / 118 = 11 (JD)
2975 / 175 = 17
294 / 14 = 21
2912 / 112 = 26
29 / 1 = 29
18292918 / 181118 = 101 (JD)
29739294 / 173914 = 171 (JD)
		297 / 27 = 11
2925 / 225 = 13
7292907 / 72207 = 101 (JD)
2972975 / 27275 = 109 (JD)
		2925 / 325 = 9
29132529 / 313253 = 93 (JD)
		129375 / 14375 = 9 (JD)
2295 / 255 = 9 (JD)
298532988 / 5853588 = 51 
		329625 / 36625 = 9 (JD)
42975 / 4775 = 9 (JD)
1295 / 185 = 7 (JD)
529875 / 58875 = 9 (JD)
		

30330 / 30 = 11
130 / 10 = 13 (JD)
3030303 / 30003 = 101 (JD)
		1309 / 119 = 11 (JD)
308125 / 18125 = 17
3045 / 145 = 21
3016 / 116 = 26
30 / 1 = 30
19303019 / 191119 = 101 (JD)
		308 / 28 = 11
30 / 2 = 15
8303008 / 82208 = 101 (JD)
		30375 / 3375 = 9
30 / 3 = 10
303303 / 3333 = 91 
		1305 / 145 = 9 (JD)
30 / 5 = 6
230625 / 25625 = 9 (JD)
		30 / 6 = 5
33075 / 3675 = 9 (JD)
		430875 / 47875 = 9 (JD)
304130825 / 7417825 = 41
430830 / 4787 = 90
6303075 / 67775 = 93
1130304 / 11774 = 96 
		30730 / 878 = 35 
23012305 / 291295 = 79 (JD)

313311 / 301 = 11 
231 / 21 = 11 (JD)
31875 / 1875 = 17
315 / 15 = 21
3125 / 125 = 25
312 / 12 = 26
31 / 1 = 31
2031312 / 20112 = 101 (JD)
		319 / 29 = 11
9313109 / 92209 = 101 (JD)
3153125 / 25225 = 125 (JD)
		315 / 35 = 9
3194316 / 39436 = 81 
		315 / 45 = 7
131625 / 14625 = 9 (JD)
31311 / 441 = 71
131313 / 1443 = 91 
		312 / 52 = 6
23175 / 2575 = 9 (JD)
		3125 / 625 = 5
331875 / 36875 = 9 (JD)
3109317 / 60967 = 51 (JD)
		317413185 / 7741785 = 41 
2315314125 / 28584125 = 81 (JD)
531 / 59 = 9 (JD)

323322 / 302 = 11 
2321 / 211 = 11 (JD)
132 / 11 = 12 (JD)
329375 / 19375 = 17
3255 / 155 = 21
3224 / 124 = 26
32 / 1 = 32 
		132 / 12 = 11 (JD)
325 / 25 = 13
32 / 2 = 16
1032321 / 10221 = 101 (JD)
3283032 / 28302 = 116 (JD)
321328 / 2128 = 151 (JD)
		32625 / 3625 = 9 
32 / 4 = 8
13275 / 1475 = 9 (JD)
		324 / 54 = 6
232875 / 25875 = 9 (JD)
232323 / 2553 = 91 
		325 / 65 = 5 
39321910112359550561325 / 397191011235955056175 = 99 (JD)
32 / 8 = 4
432 / 48 = 9 (JD)
		232 / 29 = 8 (JD)
532125 / 59125 = 9 (JD)

333333 / 303 = 11
13325 / 1025 = 13
333333 / 3003 = 111 (JD)
1333325 / 10025 = 133 (JD)
		2332 / 212 = 11 (JD)
336 / 16 = 21
3328 / 128 = 26
33 / 1 = 33 
		1331 / 121 = 11 (JD)
11333311 / 112211 = 101 (JD)
		3375 / 375 = 9
33 / 3 = 11
3333 / 33 = 101 (JD)
		133875 / 14875 = 9 (JD)
336 / 56 = 6 
3375 / 675 = 5
333333 / 3663 = 91 
		333 / 37 = 9 (JD)
433125 / 48125 = 9 (JD)
133 / 19 = 7 (JD)
53325 / 5925 = 9 (JD)
13333 / 199 = 67
5333325 / 59925 = 89 

343344 / 304 = 11 
2343 / 213 = 11 (JD)
3465 / 165 = 21
345 / 15 = 23
34375 / 1375 = 25
3432 / 132 = 26
341 / 11 = 31
3403125 / 103125 = 33 (JD)
34 / 1 = 34
6346345 / 61615 = 103 (JD)
13434025 / 111025 = 121 (JD)
		1342 / 122 = 11 (JD)
34 / 2 = 17
12343412 / 122212 = 101 (JD)
3444348 / 24428 = 141 (JD)
		34875 / 3875 = 9
341 / 31 = 11
1343401 / 13301 = 101 (JD)
		345344 / 4544 = 76 
348 / 58 = 6 
234 / 26 = 9 (JD)
83497344 / 869764 = 96 (JD)
313434 / 3166 = 99 
		334125 / 37125 = 9 (JD)
434343 / 4773 = 91 
		43425 / 4825 = 9 (JD)
534375 / 59375 = 9 (JD)

353355 / 305 = 11 
2354 / 214 = 11 (JD)
357 / 17 = 21
3536 / 136 = 26
350625 / 10625 = 33 (JD)
35 / 1 = 35 
		1353 / 123 = 11 (JD)
3575 / 275 = 13
352 / 22 = 16
350625 / 20625 = 17 (JD)
13353513 / 132213 = 101 (JD)
13581356 / 128126 = 106 (JD)
		352 / 32 = 11
2353502 / 23302 = 101 (JD)
		350789356 / 4078946 = 86 (JD)
1235831355 / 12483145 = 99 (JD)
		35 / 5 = 7
135 / 15 = 9 (JD)
35882335 / 588235 = 61 (JD)
63535 / 655 = 97 
		235125 / 26125 = 9 (JD)
35 / 7 = 5
33525 / 3725 = 9 (JD)
		435375 / 48375 = 9 (JD)
357035175 / 8708175 = 41
535353 / 5883 = 91 
		5355 / 595 = 9 (JD)

363366 / 306 = 11
1365 / 105 = 13 
		2365 / 215 = 11 (JD)
3675 / 175 = 21
364 / 14 = 26
3625 / 125 = 29
3609375 / 109375 = 33 (JD)
36 / 1 = 36 
		1364 / 124 = 11 (JD)
36125 / 2125 = 17
36 / 2 = 18
14363614 / 142214 = 101 (JD)
361536 / 2152 = 168 (JD)
		363 / 33 = 11
36 / 3 = 12
3363603 / 33303 = 101 (JD)
3623536 / 32353 = 112 (JD)
		36 / 4 = 9
36369 / 449 = 81 
		136125 / 15125 = 9 (JD)
3653364 / 55354 = 66 
		36 / 6 = 6
23625 / 2625 = 9 (JD)
		3625 / 725 = 5
136 / 17 = 8 (JD)
336375 / 37375 = 9 (JD)
36631536 / 763157 = 48 (JD)
		4365 / 485 = 9 (JD)
36367 / 887 = 41 
		36 / 9 = 4
1365 / 195 = 7 (JD)
536625 / 59625 = 9 (JD)
33653625 / 395925 = 85 (JD)
636363 / 6993 = 91 

373377 / 307 = 11 
2376 / 216 = 11 (JD)
378 / 18 = 21
375 / 15 = 25
3744 / 144 = 26
372 / 12 = 31
37125 / 1125 = 33
37 / 1 = 37 
		1375 / 125 = 11 (JD)
375 / 25 = 15
371875 / 21875 = 17
15373715 / 152215 = 101 (JD)
		374 / 34 = 11
3773764 / 37364 = 101 (JD)
		37125 / 4125 = 9 
13725 / 1525 = 9 (JD)
3724376 / 52456 = 71 (JD)
437376 / 4556 = 96 
		372 / 62 = 6
237375 / 26375 = 9 (JD)
		375 / 75 = 5
3375 / 375 = 9 (JD)
3724376 / 52456 = 71 
		437625 / 48625 = 9 (JD)
53775 / 5975 = 9 (JD)

383388 / 308 = 11 
2387 / 217 = 11 (JD)
3885 / 185 = 21
3848 / 148 = 26
3815625 / 115625 = 33
38 / 1 = 38
3896838 / 19681 = 198 (JD)
		1386 / 126 = 11 (JD)
384 / 24 = 16
3825 / 225 = 17
38 / 2 = 19
16383816 / 162216 = 101 (JD)
		385 / 35 = 11
5383805 / 53305 = 101 (JD)
		3825 / 425 = 9 
385 / 55 = 7
138375 / 15375 = 9 (JD)
		384 / 64 = 6
2385 / 265 = 9 (JD)
		3875 / 775 = 5
338625 / 37625 = 9 (JD)
		43875 / 4875 = 9 (JD)
538875 / 59875 = 9 (JD)

393399 / 309 = 11
13975 / 1075 = 13 
		2398 / 218 = 11 (JD)
399 / 19 = 21
3952 / 152 = 26
391875 / 11875 = 33 (JD)
39 / 1 = 39 
		1397 / 127 = 11 (JD)
393125 / 23125 = 17
17393917 / 172217 = 101 (JD)
3983955 / 28255 = 141 (JD)
		396 / 36 = 11
39 / 3 = 13
6393906 / 63306 = 101 (JD)
3924393 / 32433 = 121 (JD)
		39375 / 4375 = 9 
1395 / 155 = 9 (JD)
396 / 66 = 6
239625 / 26625 = 9 (JD)
3943955 / 64655 = 61 
		33975 / 3775 = 9 (JD)
393924 / 7724 = 51 
		439875 / 48875 = 9 (JD)
193939375 / 1999375 = 97 

40440 / 40 = 11
240 / 20 = 12 (JD)
140 / 10 = 14 (JD)
4040404 / 40004 = 101 (JD)
16408740 / 160870 = 102 (JD)
		2409 / 219 = 11 (JD)
4095 / 195 = 21
40625 / 1625 = 25
4056 / 156 = 26
405 / 15 = 27
403 / 13 = 31
4021875 / 121875 = 33
40 / 1 = 40
14040432 / 111432 = 126 (JD)
		1408 / 128 = 11 (JD)
40375 / 2375 = 17
40 / 2 = 20
18404018 / 182218 = 101 (JD)
404096 / 2296 = 176 (JD)
		407 / 37 = 11
7404007 / 73307 = 101 (JD)
		405 / 45 = 9
40 / 4 = 10
404404 / 4444 = 91
1404075 / 14475 = 97 
		40 / 5 = 8
140625 / 15625 = 9 (JD)
406440 / 5645 = 72
3404640 / 35465 = 96 
		408 / 68 = 6
24075 / 2675 = 9 (JD)
		340875 / 37875 = 9 (JD)
40 / 8 = 5 
50400405 / 509095 = 99 (JD)

414411 / 401 = 11 
341 / 31 = 11 (JD)
416 / 16 = 26
4125 / 125 = 33
41 / 1 = 41
3041413 / 30113 = 101 (JD)
		1419 / 129 = 11 (JD)
416 / 26 = 16
414375 / 24375 = 17
19414119 / 192219 = 101 (JD)
		418 / 38 = 11
8414108 / 83308 = 101 (JD)
		41625 / 4625 = 9
41868412 / 486842 = 86 
		14175 / 1575 = 9 (JD)
4144128 / 54528 = 76
141414 / 1554 = 91 
		241875 / 26875 = 9 (JD)
41906941 / 790697 = 53 (JD)
4125 / 825 = 5
39410415 / 398085 = 99 (JD)
		441 / 49 = 9 (JD)
12412416 / 129296 = 96 (JD)

424422 / 402 = 11
742424 / 7004 = 106 (JD)
2424235 / 20035 = 121 (JD)
		3421 / 311 = 11 (JD)
4264 / 164 = 26
4228125 / 128125 = 33
42025 / 1025 = 41 (JD)
42 / 1 = 42 
		242 / 22 = 11 (JD)
425 / 25 = 17
42 / 2 = 21
1842442 / 18242 = 101 (JD)
		429 / 39 = 11
4225 / 325 = 13
42 / 3 = 14
9424209 / 93309 = 101 (JD)
1424428 / 13438 = 106 (JD)
4242672 / 33672 = 126 (JD)
		4275 / 475 = 9 
142875 / 15875 = 9 (JD)
42 / 6 = 7
242424 / 2664 = 91 
		42 / 7 = 6
28420425 / 287075 = 99 (JD)
		425 / 85 = 5
342 / 38 = 9 (JD)
4224228 / 82828 = 51 
		442125 / 49125 = 9 (JD)

434433 / 403 = 11 
3432 / 312 = 11 (JD)
143 / 11 = 13 (JD)
4375 / 175 = 25
4368 / 168 = 26
435 / 15 = 29
434 / 14 = 31
433125 / 13125 = 33
432 / 12 = 36
4305 / 105 = 41 (JD)
43 / 1 = 43 
		2431 / 221 = 11 (JD)
435625 / 25625 = 17
4305 / 205 = 21 (JD)
		143 / 13 = 11 (JD)
1043431 / 10331 = 101 (JD)
4305435 / 30535 = 141 (JD)
		43875 / 4875 = 9 
433431 / 5351 = 81 
17430435 / 176065 = 99 (JD)
432 / 72 = 6
243 / 27 = 9 (JD)
43058436 / 705876 = 61 (JD)
14343075 / 177075 = 81 (JD)
343434 / 3774 = 91
304341543 / 3074157 = 99 
		4375 / 875 = 5
343125 / 38125 = 9 (JD)
		44325 / 4925 = 9 (JD)

444444 / 404 = 11
444444 / 4004 = 111 (JD)
		3443 / 313 = 11 (JD)
4472 / 172 = 26
4434375 / 134375 = 33
44075 / 1075 = 41 (JD)
44 / 1 = 44 
		2442 / 222 = 11 (JD)
144 / 12 = 12 (JD)
448 / 28 = 16
44625 / 2625 = 17
441 / 21 = 21
44 / 2 = 22
114444 / 1122 = 102 (JD)
14494144 / 129412 = 112 (JD)
444444 / 2442 = 182 (JD)
		1441 / 131 = 11 (JD)
11444411 / 113311 = 101 (JD)
		44 / 4 = 11
4444 / 44 = 101 (JD)
		6440445 / 65055 = 99 (JD)
144 / 16 = 9 (JD)
444 / 74 = 6
244125 / 27125 = 9 (JD)
		144 / 18 = 8 (JD)
34425 / 3825 = 9 (JD)
444444 / 4884 = 91 
		444375 / 49375 = 9 (JD)
44444375 / 499375 = 89 

454455 / 405 = 11 
3454 / 314 = 11 (JD)
4576 / 176 = 26
45375 / 1375 = 33
451 / 11 = 41
45 / 1 = 45
245456 / 2116 = 116 (JD)
		2453 / 223 = 11 (JD)
456875 / 26875 = 17
4515 / 215 = 21 
		1452 / 132 = 11 (JD)
455 / 35 = 13
45 / 3 = 15
12454512 / 123312 = 101 (JD)
		451 / 41 = 11
1454501 / 14401 = 101 (JD)
		45 / 5 = 9
94545 / 955 = 99 
		455 / 65 = 7
145125 / 16125 = 9 (JD)
		456 / 76 = 6
24525 / 2725 = 9 (JD)
		345375 / 38375 = 9 (JD)
45 / 9 = 5
4455 / 495 = 9 (JD)
4522455 / 92295 = 49
1454525 / 19925 = 73
545454 / 5994 = 91 

464466 / 406 = 11 
3465 / 315 = 11 (JD)
14625 / 1125 = 13 (JD)
46875 / 1875 = 25
468 / 18 = 26
465 / 15 = 31
4640625 / 140625 = 33
4625 / 125 = 37
46125 / 1125 = 41
46 / 1 = 46 
		2464 / 224 = 11 (JD)
4675 / 275 = 17
462 / 22 = 21
46 / 2 = 23 
		1463 / 133 = 11 (JD)
13464613 / 133313 = 101 (JD)
4683946 / 38393 = 122 (JD)
		462 / 42 = 11
2464602 / 24402 = 101 (JD)
464625 / 4425 = 105 (JD)
		46125 / 5125 = 9 
14625 / 1625 = 9 (JD)
468 / 78 = 6
246375 / 27375 = 9 (JD)
		3465 / 385 = 9 (JD)
4625 / 925 = 5
446625 / 49625 = 9 (JD)
34686465 / 398695 = 87 (JD)

474477 / 407 = 11 
3476 / 316 = 11 (JD)
4784 / 184 = 26
474375 / 14375 = 33
4715 / 115 = 41
47 / 1 = 47 
		2475 / 225 = 11 (JD)
478125 / 28125 = 17
475 / 25 = 19
4725 / 225 = 21
4754475 / 25425 = 187 (JD)
		1474 / 134 = 11 (JD)
14474714 / 143314 = 101 (JD)
		473 / 43 = 11
3474703 / 34403 = 101 (JD)
		4725 / 525 = 9
4791474 / 59154 = 81
47472 / 552 = 86
24747125 / 255125 = 97 
		147375 / 16375 = 9 (JD)
2475 / 275 = 9 (JD)
347625 / 38625 = 9 (JD)
475 / 95 = 5
44775 / 4975 = 9 (JD)

484488 / 408 = 11 
3487 / 317 = 11 (JD)
4888 / 188 = 26
4846875 / 146875 = 33
48175 / 1175 = 41
48 / 1 = 48
4485348 / 41531 = 108 (JD)
2484488 / 21418 = 116 (JD)
		2486 / 226 = 11 (JD)
48875 / 2875 = 17
483 / 23 = 21
48 / 2 = 24 
		1485 / 135 = 11 (JD)
4875 / 375 = 13
48 / 3 = 16
15484815 / 153315 = 101 (JD)
14864875 / 136375 = 109 (JD)
		484 / 44 = 11
48 / 4 = 12
4484804 / 44404 = 101 (JD)
		48375 / 5375 = 9 
48 / 6 = 8
1485 / 165 = 9 (JD)
4834816 / 63616 = 76 
		248625 / 27625 = 9 (JD)
64855488 / 675578 = 96 (JD)
		48 / 8 = 6
34875 / 3875 = 9 (JD)
94848 / 988 = 96 
		4875 / 975 = 5
448875 / 49875 = 9 (JD)
4844898 / 94998 = 51 

494499 / 409 = 11 
3498 / 318 = 11 (JD)
1495 / 115 = 13 (JD)
4992 / 192 = 26
496 / 16 = 31
495 / 15 = 33
492 / 12 = 41
49 / 1 = 49 
		2497 / 227 = 11 (JD)
499375 / 29375 = 17
4935 / 235 = 21
499491 / 2921 = 171 (JD)
		1496 / 136 = 11 (JD)
16494916 / 163316 = 101 (JD)
		495 / 45 = 11
4994955 / 49455 = 101 (JD)
		495 / 55 = 9 
149625 / 16625 = 9 (JD)
49 / 7 = 7
24975 / 2775 = 9 (JD)
		492 / 82 = 6
349875 / 38875 = 9 (JD)
		49755498 / 975598 = 51 

50550 / 50 = 11
150 / 10 = 15 (JD)
5050505 / 50005 = 101 (JD)
		3509 / 319 = 11 (JD)
5096 / 196 = 26
5075 / 175 = 29
5053125 / 153125 = 33
504 / 14 = 36
50225 / 1225 = 41
50 / 1 = 50 
		2508 / 228 = 11 (JD)
504 / 24 = 21
50 / 2 = 25 
		1507 / 137 = 11 (JD)
17505017 / 173317 = 101 (JD)
505505 / 3535 = 143 (JD)
		506 / 46 = 11
6505006 / 64406 = 101 (JD)
		50625 / 5625 = 9
50 / 5 = 10
506250 / 5625 = 90
505505 / 5555 = 91 
		15075 / 1675 = 9 (JD)
505505 / 6565 = 77
501950925 / 6196925 = 81 
		250875 / 27875 = 9 (JD)
504 / 84 = 6
15055065 / 185865 = 81 (JD)
		505002 / 9902 = 51 (JD)
4850505 / 48995 = 99 

515511 / 501 = 11
1751515 / 17005 = 103 (JD)
1516515 / 10605 = 143 (JD)
1515134 / 10034 = 151 (JD)
		451 / 41 = 11 (JD)
515625 / 15625 = 33
5125 / 125 = 41
51 / 1 = 51
4051514 / 40114 = 101 (JD)
		2519 / 229 = 11 (JD)
5145 / 245 = 21 
		1518 / 138 = 11 (JD)
512 / 32 = 16
51 / 3 = 17
18515118 / 183318 = 101 (JD)
		517 / 47 = 11
7515107 / 74407 = 101 (JD)
		5175 / 575 = 9
5105146 / 50546 = 101 (JD)
		151875 / 16875 = 9 (JD)
51715125 / 671625 = 77
151515 / 1665 = 91 
		51213114512 / 721311472 = 71 (JD)
651906976512 / 6790697672 = 96 (JD)
		516 / 86 = 6
37519515 / 378985 = 99 (JD)
		351 / 39 = 9 (JD)
1515136 / 19936 = 76
1516515 / 19695 = 77 (JD)
1551515 / 15995 = 97 

525522 / 502 = 11 
4521 / 411 = 11 (JD)
252 / 21 = 12 (JD)
15275 / 1175 = 13 (JD)
527 / 17 = 31
5259375 / 159375 = 33
525 / 15 = 35
52275 / 1275 = 41
520625 / 10625 = 49 (JD)
5202 / 102 = 51 (JD)
52 / 1 = 52
3524752 / 31471 = 112 (JD)
		352 / 32 = 11 (JD)
525 / 25 = 21
52 / 2 = 26
3052523 / 30223 = 101 (JD)
		1529 / 139 = 11 (JD)
525 / 35 = 15
520625 / 30625 = 17 (JD)
19525219 / 193319 = 101 (JD)
		528 / 48 = 11
52 / 4 = 13
8525208 / 84408 = 101 (JD)
152852 / 1484 = 103 (JD)
5232524 / 43244 = 121 (JD)
		52875 / 5875 = 9
15205247 / 150547 = 101 (JD)
		2522790528 / 26279068 = 96 (JD)
525 / 75 = 7
252525 / 2775 = 91
26529525 / 267975 = 99 (JD)
		528 / 88 = 6
252 / 28 = 9 (JD)
152523 / 1883 = 81 
		152 / 19 = 8 (JD)
352125 / 39125 = 9 (JD)

535533 / 503 = 11 
4532 / 412 = 11 (JD)
53625 / 1625 = 33
533 / 13 = 41
5304 / 104 = 51 (JD)
53 / 1 = 53 
		3531 / 321 = 11 (JD)
5355 / 255 = 21
53125 / 2125 = 25
5304 / 204 = 26 (JD)
553532 / 5222 = 106 (JD)
		253 / 23 = 11 (JD)
53125 / 3125 = 17
2053532 / 20332 = 101 (JD)
535325 / 3325 = 161 (JD)
		539 / 49 = 11
9535309 / 94409 = 101 (JD)
		53531875 / 551875 = 97 (JD)
53530875 / 660875 = 81 (JD)
15539535 / 156965 = 99 (JD)
		153 / 17 = 9 (JD)
253125 / 28125 = 9 (JD)
353535 / 3885 = 91 
		35325 / 3925 = 9 (JD)

545544 / 504 = 11
654545 / 6005 = 109 (JD)
		4543 / 413 = 11 (JD)
154 / 11 = 14 (JD)
5465625 / 165625 = 33
54325 / 1325 = 41
5406 / 106 = 51 (JD)
54 / 1 = 54
35454 / 311 = 114 (JD)
		3542 / 322 = 11 (JD)
546 / 26 = 21
5408 / 208 = 26 (JD)
54 / 2 = 27
15454656 / 122656 = 126 (JD)
		2541 / 231 = 11 (JD)
544 / 34 = 16
541875 / 31875 = 17 (JD)
54 / 3 = 18 
		154 / 14 = 11 (JD)
1054541 / 10441 = 101 (JD)
		4549545 / 45955 = 99 (JD)
35405449 / 350549 = 101 (JD)
		54 / 6 = 9
15404544 / 160464 = 96 (JD)
		154125 / 17125 = 9 (JD)
25425 / 2825 = 9 (JD)
54 / 9 = 6
354375 / 39375 = 9 (JD)
454545 / 4995 = 91 

555555 / 505 = 11
555555 / 5005 = 111 (JD)
		4554 / 414 = 11 (JD)
558 / 18 = 31
556875 / 16875 = 33
555 / 15 = 37
5535 / 135 = 41
552 / 12 = 46
55125 / 1125 = 49
5508 / 108 = 51 (JD)
55 / 1 = 55
255552 / 2112 = 121 (JD)
		3553 / 323 = 11 (JD)
5565 / 265 = 21
5512 / 212 = 26 
		2552 / 232 = 11 (JD)
5525 / 325 = 17 
		1551 / 141 = 11 (JD)
5525 / 425 = 13
11555511 / 114411 = 101 (JD)
5555088 / 44088 = 126 (JD)
		55 / 5 = 11
5555 / 55 = 101 (JD)
		55125 / 6125 = 9
255552 / 2662 = 96 
		15525 / 1725 = 9 (JD)
5525504 / 72704 = 76 (JD)
		255375 / 28375 = 9 (JD)
552 / 92 = 6
3555 / 395 = 9 (JD)
55552 / 992 = 56
355555 / 3995 = 89
7555255 / 79529 = 95 

565566 / 506 = 11 
4565 / 415 = 11 (JD)
5671875 / 171875 = 33
56375 / 1375 = 41
5625 / 125 = 45
561 / 11 = 51
56 / 1 = 56 
		3564 / 324 = 11 (JD)
156 / 12 = 13 (JD)
567 / 27 = 21
5625 / 225 = 25
5616 / 216 = 26
56 / 2 = 28 
		2563 / 233 = 11 (JD)
156 / 13 = 12 (JD)
563125 / 33125 = 17 
		1562 / 142 = 11 (JD)
56 / 4 = 14
12565612 / 124412 = 101 (JD)
5656896 / 44896 = 126 (JD)
		561 / 51 = 11
1565601 / 15501 = 101 (JD)
		5625 / 625 = 9 
56 / 7 = 8
156375 / 17375 = 9 (JD)
		56 / 8 = 7
2565 / 285 = 9 (JD)
		564 / 94 = 6
356625 / 39625 = 9 (JD)

575577 / 507 = 11
1575 / 105 = 15 
		4576 / 416 = 11 (JD)
5775 / 175 = 33
576 / 16 = 36
574 / 14 = 41
5712 / 112 = 51
57 / 1 = 57
15749657 / 114961 = 137 (JD)
		3575 / 325 = 11 (JD)
5775 / 275 = 21
575 / 25 = 23
572 / 22 = 26 
		2574 / 234 = 11 (JD)
576 / 36 = 16
57375 / 3375 = 17
57 / 3 = 19
5785716 / 38316 = 151 (JD)
		1573 / 143 = 11 (JD)
13575713 / 134413 = 101 (JD)
		572 / 52 = 11
2575702 / 25502 = 101 (JD)
		57375 / 6375 = 9 
1575 / 175 = 9 (JD)
257625 / 28625 = 9 (JD)
576 / 96 = 6
35775 / 3975 = 9 (JD)

585588 / 508 = 11 
4587 / 417 = 11 (JD)
589 / 19 = 31
5878125 / 178125 = 33
585 / 15 = 39
58425 / 1425 = 41
581875 / 11875 = 49
5814 / 114 = 51
58 / 1 = 58 
		3586 / 326 = 11 (JD)
588 / 28 = 21
5824 / 224 = 26
58 / 2 = 29 
		2585 / 235 = 11 (JD)
584375 / 34375 = 17
5831584 / 33134 = 176 (JD)
		1584 / 144 = 11 (JD)
585 / 45 = 13
14585814 / 144414 = 101 (JD)
5861558 / 46154 = 127 (JD)
		583 / 53 = 11
3585803 / 35503 = 101 (JD)
		585 / 65 = 9
58265058 / 626506 = 93 (JD)
		158625 / 17625 = 9 (JD)
585808 / 7708 = 76 (JD)
		25875 / 2875 = 9 (JD)
588 / 98 = 6
358875 / 39875 = 9 (JD)

595599 / 509 = 11 
4598 / 418 = 11 (JD)
598125 / 18125 = 33
5945 / 145 = 41
5916 / 116 = 51
59 / 1 = 59 
		3597 / 327 = 11 (JD)
15925 / 1225 = 13 (JD)
5985 / 285 = 21
59375 / 2375 = 25
5928 / 228 = 26 
		2596 / 236 = 11 (JD)
595 / 35 = 17 
		1595 / 145 = 11 (JD)
15595915 / 154415 = 101 (JD)
5975928 / 47428 = 126 (JD)
		594 / 54 = 11
4595904 / 45504 = 101 (JD)
594595 / 5455 = 109 (JD)
		59625 / 6625 = 9 
15975 / 1775 = 9 (JD)
595 / 85 = 7
259875 / 28875 = 9 (JD)
		1597183059375 / 19718309375 = 81 (JD)

60660 / 60 = 11
360 / 30 = 12 (JD)
260 / 20 = 13 (JD)
160 / 10 = 16 (JD)
6060606 / 60006 = 101 (JD)
1608760 / 10870 = 148 (JD)
		4609 / 419 = 11 (JD)
6084375 / 184375 = 33
60475 / 1475 = 41
6018 / 118 = 51
60 / 1 = 60 
		3608 / 328 = 11 (JD)
609 / 29 = 21
6032 / 232 = 26
60 / 2 = 30
160606 / 1226 = 131 (JD)
		2607 / 237 = 11 (JD)
608 / 38 = 16
605625 / 35625 = 17
60 / 3 = 20 
		1606 / 146 = 11 (JD)
60 / 4 = 15
16606016 / 164416 = 101 (JD)
6027609 / 42749 = 141 (JD)
		605 / 55 = 11
60 / 5 = 12
5606005 / 55505 = 101 (JD)
		6075 / 675 = 9
60 / 6 = 10
606606 / 6666 = 91
26089605 / 268965 = 97 (JD)
		160875 / 17875 = 9 (JD)
60560 / 757 = 80 
		605245606 / 8524586 = 71 (JD)
4604705860 / 48470588 = 95 (JD)
		606035 / 9935 = 61 

616611 / 601 = 11 
561 / 51 = 11 (JD)
61875 / 1875 = 33
615 / 15 = 41
6125 / 125 = 49
612 / 12 = 51
61 / 1 = 61
5061615 / 50115 = 101 (JD)
		3619 / 329 = 11 (JD)
6195 / 295 = 21
6136 / 236 = 26 
		2618 / 238 = 11 (JD)
61625 / 3625 = 17 (JD)
		1617 / 147 = 11 (JD)
6175 / 475 = 13
17616117 / 174417 = 101 (JD)
261615 / 2445 = 107 (JD)
6106138 / 40438 = 151 (JD)
		616 / 56 = 11
6616106 / 65506 = 101 (JD)
		61875 / 6875 = 9
616125 / 6625 = 93 
		161616 / 1776 = 91 
2661615 / 26885 = 99 
261 / 29 = 9 (JD)
16161525 / 199525 = 81 

626622 / 602 = 11 
5621 / 511 = 11 (JD)
6290625 / 190625 = 33
62525 / 1525 = 41
6222 / 122 = 51
62 / 1 = 62 
		462 / 42 = 11 (JD)
1625 / 125 = 13 (JD)
625 / 25 = 25
624 / 24 = 26
62 / 2 = 31
4062624 / 40224 = 101 (JD)
		2629 / 239 = 11 (JD)
626875 / 36875 = 17 
		1628 / 148 = 11 (JD)
18626218 / 184418 = 101 (JD)
		627 / 57 = 11
7626207 / 75507 = 101 (JD)
		62153846216 / 615384616 = 101 (JD)
62623125 / 773125 = 81 (JD)
162 / 18 = 9 (JD)
62622 / 882 = 71
262626 / 2886 = 91 
		262125 / 29125 = 9 (JD)

636633 / 603 = 11
26325 / 2025 = 13
1632 / 102 = 16 
		5632 / 512 = 11 (JD)
639375 / 19375 = 33
6355 / 155 = 41
6324 / 124 = 51
3 / 1 = 63 
		4631 / 421 = 11 (JD)
6344 / 244 = 26 
		363 / 33 = 11 (JD)
6375 / 375 = 17
63 / 3 = 21
2763663 / 27363 = 101 (JD)
		1639 / 149 = 11 (JD)
19636319 / 194419 = 101 (JD)
63635 / 445 = 143 (JD)
		638 / 58 = 11
8636308 / 85508 = 101 (JD)
		163896353125 / 1689653125 = 97 (JD)
6326337 / 62637 = 101 (JD)
		63 / 7 = 9
6388632 / 78872 = 81 
		163125 / 18125 = 9 (JD)
63 / 9 = 7
26325 / 2925 = 9 (JD)
363636 / 3996 = 91
116363625 / 1199625 = 97 

646644 / 604 = 11 
5643 / 513 = 11 (JD)
6496875 / 196875 = 33
648 / 18 = 36
6475 / 175 = 37
64575 / 1575 = 41
645 / 15 = 43
644 / 14 = 46
643125 / 13125 = 49
6426 / 126 = 51
64125 / 1125 = 57
6405 / 105 = 61 (JD)
64 / 1 = 64 
		4642 / 422 = 11 (JD)
264 / 22 = 12 (JD)
6448 / 248 = 26
64 / 2 = 32 
		3641 / 331 = 11 (JD)
648125 / 38125 = 17
6405 / 305 = 21 (JD)
		264 / 24 = 11 (JD)
64 / 4 = 16
2064642 / 20442 = 101 (JD)
36464 / 344 = 106 (JD)
		649 / 59 = 11
9646409 / 95509 = 101 (JD)
646425 / 5525 = 117 (JD)
		16426438 / 162638 = 101 (JD)
641702127664 / 6170212766 = 104 (JD)
		64125 / 7125 = 9 
64 / 8 = 8
16425 / 1825 = 9 (JD)
864164 / 8818 = 98 
		264375 / 29375 = 9 (JD)

656655 / 605 = 11 
5654 / 514 = 11 (JD)
165 / 11 = 15 (JD)
656 / 16 = 41
6528 / 128 = 51
65 / 1 = 65 
		4653 / 423 = 11 (JD)
16575 / 1275 = 13 (JD)
65625 / 2625 = 25
6552 / 252 = 26
6525 / 225 = 29
651 / 21 = 31 
		3652 / 332 = 11 (JD)
65875 / 3875 = 17
651 / 31 = 21 
		2651 / 241 = 11 (JD)
165 / 15 = 11 (JD)
65 / 5 = 13
1065651 / 10551 = 101 (JD)
36565 / 355 = 103 (JD)
6540655 / 54055 = 121 (JD)
		26526539 / 262639 = 101 (JD)
6525 / 725 = 9 
165375 / 18375 = 9 (JD)
4654656 / 48486 = 96 (JD)
		2655 / 295 = 9 (JD)
6517658 / 91798 = 71 

666666 / 606 = 11
2665 / 205 = 13
1664 / 104 = 16
666666 / 6006 = 111 (JD)
266665 / 2005 = 133 (JD)
166664 / 1004 = 166 (JD)
		5665 / 515 = 11 (JD)
66625 / 1625 = 41
663 / 13 = 51
6625 / 125 = 53
66015625 / 1015625 = 65 (JD)
66 / 1 = 66
2666664 / 21164 = 126 (JD)
16639766 / 113971 = 146 (JD)
		4664 / 424 = 11 (JD)
6656 / 256 = 26
66 / 2 = 33 
		3663 / 333 = 11 (JD)
669375 / 39375 = 17
665 / 35 = 19
6615 / 315 = 21
66 / 3 = 22
666666 / 3663 = 182 (JD)
		2662 / 242 = 11 (JD)
1661 / 151 = 11 (JD)
11666611 / 115511 = 101 (JD)
		66 / 6 = 11
6666 / 66 = 101 (JD)
		66375 / 7375 = 9 
1665 / 185 = 9 (JD)
661666725 / 8168725 = 81 
		665 / 95 = 7
266625 / 29625 = 9 (JD)
66665 / 995 = 67
26666625 / 299625 = 89 

676677 / 607 = 11 
5676 / 516 = 11 (JD)
6765 / 165 = 41
675 / 15 = 45
67375 / 1375 = 49
6732 / 132 = 51
672 / 12 = 56
671 / 11 = 61
6703125 / 103125 = 65 (JD)
67 / 1 = 67
1676704 / 11104 = 151 (JD)
		4675 / 425 = 11 (JD)
676 / 26 = 26
675 / 25 = 27
6703125 / 203125 = 33 (JD)
		3674 / 334 = 11 (JD)
672 / 32 = 21
16767072 / 133072 = 126 (JD)
		2673 / 243 = 11 (JD)
675 / 45 = 15
672 / 42 = 16
16743672 / 144342 = 116 (JD)
676782 / 4482 = 151 (JD)
		1661 / 151 = 11 (JD)
12676712 / 125512 = 101 (JD)
		671 / 61 = 11
1676701 / 16601 = 101 (JD)
		675 / 75 = 9 
167625 / 18625 = 9 (JD)
26775 / 2975 = 9 (JD)

686688 / 608 = 11 
5687 / 517 = 11 (JD)
68675 / 1675 = 41
6834 / 134 = 51
68046875 / 1046875 = 65 (JD)
68 / 1 = 68 
		4686 / 426 = 11 (JD)
168 / 12 = 14 (JD)
6875 / 275 = 25
6864 / 264 = 26
682 / 22 = 31
680625 / 20625 = 33 (JD)
68 / 2 = 34
2686805 / 22205 = 121 (JD)
		3685 / 335 = 11 (JD)
6825 / 325 = 21
1686888 / 13388 = 126 (JD)
		2684 / 244 = 11 (JD)
168 / 14 = 12 (JD)
68 / 4 = 17 
		1683 / 153 = 11 (JD)
6825 / 525 = 13
13686813 / 135513 = 101 (JD)
		682 / 62 = 11
2686802 / 26602 = 101 (JD)
		68625 / 7625 = 9 
16875 / 1875 = 9 (JD)
268875 / 29875 = 9 (JD)

696699 / 609 = 11
26975 / 2075 = 13
1696 / 106 = 16 
		5698 / 518 = 11 (JD)
697 / 17 = 41
6936 / 136 = 51
690625 / 10625 = 65 (JD)
69 / 1 = 69 
		4697 / 427 = 11 (JD)
6968 / 268 = 26
6909375 / 209375 = 33 (JD)
		3696 / 336 = 11 (JD)
169 / 13 = 13 (JD)
693 / 33 = 21
69 / 3 = 23
1269269 / 12323 = 103 (JD)
		2695 / 245 = 11 (JD)
690625 / 40625 = 17 (JD)
		1694 / 154 = 11 (JD)
14696914 / 145514 = 101 (JD)
6969312 / 55312 = 126 (JD)
		693 / 63 = 11
3696903 / 36603 = 101 (JD)
6969375 / 66375 = 105 (JD)
		6975 / 775 = 9 
169875 / 18875 = 9 (JD)
969696875 / 9996875 = 97 (JD)

70770 / 70 = 11
170 / 10 = 17 (JD)
7070707 / 70007 = 101 (JD)
		5709 / 519 = 11 (JD)
70725 / 1725 = 41
705 / 15 = 47
704375 / 14375 = 49
7038 / 138 = 51
7015 / 115 = 61
70078125 / 1078125 = 65 (JD)
70 / 1 = 70 
		4708 / 428 = 11 (JD)
7072 / 272 = 26
70125 / 2125 = 33
70 / 2 = 35 
		3707 / 337 = 11 (JD)
7035 / 335 = 21 
		2706 / 246 = 11 (JD)
704 / 44 = 16
70125 / 4125 = 17
17005703 / 140543 = 121 (JD)
		1705 / 155 = 11 (JD)
70 / 5 = 14
15707015 / 155515 = 101 (JD)
		704 / 64 = 11
4707004 / 46604 = 101 (JD)
		70875 / 7875 = 9
70 / 7 = 10
707707 / 7777 = 91 
		7073239436619717015 / 87323943661971815 = 81 (JD)
370877025 / 3987925 = 93 (JD)

717711 / 701 = 11
1710625 / 100625 = 17 (JD)
		671 / 61 = 11 (JD)
7175 / 175 = 41
714 / 14 = 51
7125 / 125 = 57
7109375 / 109375 = 65 (JD)
71 / 1 = 71
6071716 / 60116 = 101 (JD)
		4719 / 429 = 11 (JD)
71875 / 2875 = 25
7176 / 276 = 26
713 / 23 = 31
7115625 / 215625 = 33 
		3718 / 338 = 11 (JD)
714 / 34 = 21 
		2717 / 247 = 11 (JD)
711875 / 41875 = 17
3717715 / 34745 = 107 (JD)
		1716 / 156 = 11 (JD)
715 / 55 = 13
16717116 / 165516 = 101 (JD)
		715 / 65 = 11
5717105 / 56605 = 101 (JD)
		71032967104 / 703296704 = 101 (JD)
71715375 / 885375 = 81 (JD)
171717 / 1887 = 91 
		171 / 19 = 9 (JD)
247166271 / 2496629 = 99 

727722 / 702 = 11
1728 / 108 = 16
172125 / 10125 = 17 
		6721 / 611 = 11 (JD)
372 / 31 = 12 (JD)
1725 / 115 = 15 (JD)
72775 / 1775 = 41
7242 / 142 = 51
72109375 / 1109375 = 65 (JD)
72 / 1 = 72 
		572 / 52 = 11 (JD)
728 / 28 = 26
725 / 25 = 29
721875 / 21875 = 33
72 / 2 = 36
5072725 / 50225 = 101 (JD)
172725 / 1225 = 141 (JD)
		3729 / 339 = 11 (JD)
17225 / 1325 = 13 (JD)
7245 / 345 = 21
72 / 3 = 24 
		2728 / 248 = 11 (JD)
7225 / 425 = 17
72 / 4 = 18
723072 / 4304 = 168 (JD)
		1727 / 157 = 11 (JD)
17727217 / 175517 = 101 (JD)
		726 / 66 = 11
72 / 6 = 12
6727206 / 66606 = 101 (JD)
7247072 / 64706 = 112 (JD)
		472725 / 4775 = 99 
72 / 8 = 9 
72 / 9 = 8
172125 / 19125 = 9 (JD)
7272625 / 99625 = 73
272727 / 2997 = 91 

737733 / 703 = 11
1731875 / 101875 = 17 (JD)
		6732 / 612 = 11 (JD)
273 / 21 = 13 (JD)
738 / 18 = 41
736 / 16 = 46
735 / 15 = 49
7344 / 144 = 51
732 / 12 = 61
73125 / 1125 = 65
73 / 1 = 73
1739736 / 11916 = 146 (JD)
		5731 / 521 = 11 (JD)
7384 / 284 = 26
7321875 / 221875 = 33 
		473 / 43 = 11 (JD)
735 / 35 = 21
4073734 / 40334 = 101 (JD)
		2739 / 249 = 11 (JD)
736 / 46 = 16
733125 / 43125 = 17 
		1738 / 158 = 11 (JD)
18737318 / 185518 = 101 (JD)
		737 / 67 = 11
7737307 / 76607 = 101 (JD)
		732527346 / 7252746 = 101 (JD)
73125 / 8125 = 9 
17325 / 1925 = 9 (JD)
2733735 / 29395 = 93 (JD)

747744 / 704 = 11
17425 / 1025 = 17 
		6743 / 613 = 11 (JD)
74825 / 1825 = 41
7446 / 146 = 51
74140625 / 1140625 = 65 (JD)
74 / 1 = 74 
		5742 / 522 = 11 (JD)
7488 / 288 = 26
744 / 24 = 31
7425 / 225 = 33
74 / 2 = 37 
		4741 / 431 = 11 (JD)
7455 / 355 = 21 
		374 / 34 = 11 (JD)
74375 / 4375 = 17
3074743 / 30443 = 101 (JD)
		1749 / 159 = 11 (JD)
7475 / 575 = 13
19747419 / 195519 = 101 (JD)
		748 / 68 = 11
8747408 / 86608 = 101 (JD)
		1742527447 / 17252747 = 101 (JD)
7428260874 / 72826087 = 102 (JD)
		7425 / 825 = 9 
174375 / 19375 = 9 (JD)

757755 / 705 = 11
1753125 / 103125 = 17 (JD)
		6754 / 614 = 11 (JD)
7585 / 185 = 41
7548 / 148 = 51
7515625 / 115625 = 65
75 / 1 = 75 
		5753 / 523 = 11 (JD)
7592 / 292 = 26
7528125 / 228125 = 33 
		4752 / 432 = 11 (JD)
1755 / 135 = 13 (JD)
756 / 36 = 21
75 / 3 = 25 
		3751 / 341 = 11 (JD)
754375 / 44375 = 17
75753 / 443 = 171 (JD)
		275 / 25 = 11 (JD)
75 / 5 = 15
2075752 / 20552 = 101 (JD)
		759 / 69 = 11
9757509 / 96609 = 101 (JD)
		7547528 / 74728 = 101 (JD)
75375 / 8375 = 9 
1755 / 195 = 9 (JD)
757568 / 9968 = 76
75275325 / 929325 = 81
7757575 / 79975 = 97 

767766 / 706 = 11
176375 / 10375 = 17 
		6765 / 615 = 11 (JD)
27625 / 2125 = 13 (JD)
176 / 11 = 16 (JD)
76875 / 1875 = 41
765625 / 15625 = 49
765 / 15 = 51
7625 / 125 = 61
76171875 / 1171875 = 65 (JD)
76 / 1 = 76
47676 / 411 = 116 (JD)
		5764 / 524 = 11 (JD)
7696 / 296 = 26
763125 / 23125 = 33
76 / 2 = 38 
		4763 / 433 = 11 (JD)
276 / 23 = 12 (JD)
7665 / 365 = 21 
		3762 / 342 = 11 (JD)
768 / 48 = 16
765 / 45 = 17
76 / 4 = 19
76176 / 414 = 184 (JD)
		2761 / 251 = 11 (JD)
176 / 16 = 11 (JD)
1076761 / 10661 = 101 (JD)
		17647629 / 174729 = 101 (JD)
762631576875 / 7263157875 = 105 (JD)
		765 / 85 = 9 
176625 / 19625 = 9 (JD)

777777 / 707 = 11
1774375 / 104375 = 17 (JD)
777777 / 7007 = 111 (JD)
		6776 / 616 = 11 (JD)
779 / 19 = 41
7752 / 152 = 51
771875 / 11875 = 65
77 / 1 = 77 
		5775 / 525 = 11 (JD)
775 / 25 = 31
7734375 / 234375 = 33 
		4774 / 434 = 11 (JD)
777 / 37 = 21 
		3773 / 343 = 11 (JD)
775625 / 45625 = 17 
		2772 / 252 = 11 (JD)
1771 / 161 = 11 (JD)
11777711 / 116611 = 101 (JD)
		77 / 7 = 11
7777 / 77 = 101 (JD)
		77625 / 8625 = 9 
17775 / 1975 = 9 (JD)
1777775 / 19975 = 89 

787788 / 708 = 11
1785 / 105 = 17 
		6787 / 617 = 11 (JD)
78925 / 1925 = 41
7875 / 175 = 45
7854 / 154 = 51
784 / 14 = 56
78375 / 1375 = 57
78203125 / 1203125 = 65 (JD)
781 / 11 = 71
78 / 1 = 78 
		5786 / 526 = 11 (JD)
78375 / 2375 = 33
78 / 2 = 39 
		4785 / 435 = 11 (JD)
17875 / 1375 = 13 (JD)
7875 / 375 = 21
78125 / 3125 = 25
78 / 3 = 26 
		3784 / 344 = 11 (JD)
78625 / 4625 = 17
7871378 / 47134 = 167 (JD)
		2783 / 253 = 11 (JD)
1782 / 162 = 11 (JD)
78 / 6 = 13
12787812 / 126612 = 101 (JD)
1783785 / 16365 = 109 (JD)
7848786 / 64866 = 121 (JD)
		781 / 71 = 11
1787801 / 17701 = 101 (JD)
		7875 / 875 = 9 
178875 / 19875 = 9 (JD)
7817875 / 91975 = 85 

797799 / 709 = 11
1795625 / 105625 = 17 (JD)
		6798 / 618 = 11 (JD)
2795 / 215 = 13 (JD)
1792 / 112 = 16 (JD)
7995 / 195 = 41
79625 / 1625 = 49
7956 / 156 = 51
795 / 15 = 53
793 / 13 = 61
7921875 / 121875 = 65
792 / 12 = 66
7904 / 104 = 76 (JD)
79 / 1 = 79
17997992 / 119192 = 151 (JD)
17979675 / 111675 = 161 (JD)
		5797 / 527 = 11 (JD)
7975 / 275 = 29
7940625 / 240625 = 33
792 / 22 = 36 
		4796 / 436 = 11 (JD)
798 / 38 = 21
7904 / 304 = 26 (JD)
		3795 / 345 = 11 (JD)
796875 / 46875 = 17 
		2794 / 254 = 11 (JD)
796791 / 5651 = 141 (JD)
		1793 / 163 = 11 (JD)
13797913 / 136613 = 101 (JD)
		792 / 72 = 11
2797902 / 27702 = 101 (JD)
		79875 / 8875 = 9 
798579 / 9859 = 81 

80880 / 80 = 11
480 / 40 = 12 (JD)
280 / 20 = 14 (JD)
180625 / 10625 = 17
180 / 10 = 18 (JD)
8080808 / 80008 = 101 (JD)
1807680 / 10760 = 168 (JD)
		6809 / 619 = 11 (JD)
80975 / 1975 = 41
8058 / 158 = 51
80234375 / 1234375 = 65 (JD)
80 / 1 = 80 
		5808 / 528 = 11 (JD)
180 / 12 = 15 (JD)
806 / 26 = 31
804375 / 24375 = 33
80 / 2 = 40 
		4807 / 437 = 11 (JD)
8085 / 385 = 21
805 / 35 = 23
8008 / 308 = 26 (JD)
		3806 / 346 = 11 (JD)
8075 / 475 = 17
80 / 4 = 20 
		2805 / 255 = 11 (JD)
180 / 15 = 12 (JD)
80 / 5 = 16
804380 / 5435 = 148 (JD)
		1804 / 164 = 11 (JD)
14808014 / 146614 = 101 (JD)
		803 / 73 = 11
3808003 / 37703 = 101 (JD)
		80 / 8 = 10
808808 / 8888 = 91 
		80807625 / 997625 = 81 (JD)

818811 / 801 = 11
1816875 / 106875 = 17 (JD)
9818175 / 90075 = 109 (JD)
1818145 / 10045 = 181 (JD)
		781 / 71 = 11 (JD)
816 / 16 = 51
8125 / 125 = 65
81 / 1 = 81
7081817 / 70117 = 101 (JD)
14818125 / 141125 = 105 (JD)
		5819 / 529 = 11 (JD)
8146875 / 246875 = 33 
		4818 / 438 = 11 (JD)
819 / 39 = 21
8125 / 325 = 25
8112 / 312 = 26
81 / 3 = 27 
		3817 / 347 = 11 (JD)
818125 / 48125 = 17
18181296 / 144296 = 126 (JD)
		2816 / 256 = 11 (JD)
1815 / 165 = 11 (JD)
8125 / 625 = 13
15818115 / 156615 = 101 (JD)
8148168 / 64668 = 126 (JD)
		814 / 74 = 11
4818104 / 47704 = 101 (JD)
		4816516815 / 48651685 = 99 (JD)
81 / 9 = 9
181818 / 1998 = 91 

828822 / 802 = 11
18275 / 1075 = 17 
		7821 / 711 = 11 (JD)
28275 / 2175 = 13 (JD)
1824 / 114 = 16 (JD)
828 / 18 = 46
826875 / 16875 = 49
8262 / 162 = 51
825 / 15 = 55
8235 / 135 = 61
82265625 / 1265625 = 65 (JD)
82125 / 1125 = 73
8208 / 108 = 76 (JD)
820125 / 10125 = 81 (JD)
82 / 1 = 82 
		682 / 62 = 11 (JD)
825 / 25 = 33
82 / 2 = 41
6082826 / 60226 = 101 (JD)
		4829 / 439 = 11 (JD)
182 / 13 = 14 (JD)
8295 / 395 = 21
8216 / 316 = 26 
		3828 / 348 = 11 (JD)
182 / 14 = 13 (JD)
82875 / 4875 = 17
13821982 / 134194 = 103 (JD)
		2827 / 257 = 11 (JD)
825 / 55 = 15 
		1826 / 166 = 11 (JD)
16828216 / 166616 = 101 (JD)
		825 / 75 = 11
5828205 / 57705 = 101 (JD)
182825 / 1775 = 103 (JD)
		821318264 / 8131864 = 101 (JD)
82125 / 9125 = 9
6826582 / 69659 = 98 

838833 / 803 = 11
1838125 / 108125 = 17 (JD)
		7832 / 712 = 11 (JD)
8364 / 164 = 51
8328125 / 128125 = 65
83025 / 1025 = 81 (JD)
83 / 1 = 83
18339832 / 113912 = 161 (JD)
		6831 / 621 = 11 (JD)
837 / 27 = 31
8353125 / 253125 = 33
8325 / 225 = 37
83025 / 2025 = 41 (JD)
		583 / 53 = 11 (JD)
832 / 32 = 26
5083835 / 50335 = 101 (JD)
		3839 / 349 = 11 (JD)
839375 / 49375 = 17 
		2838 / 258 = 11 (JD)
832 / 52 = 16
1838832 / 15852 = 116 (JD)
18313834 / 151354 = 121 (JD)
838352 / 5552 = 151 (JD)
		1837 / 167 = 11 (JD)
17838317 / 176617 = 101 (JD)
8383536 / 66536 = 126 (JD)
		836 / 76 = 11
6838306 / 67706 = 101 (JD)
		8362068390625 / 86206890625 = 97 (JD)
1831318365 / 18131865 = 101 (JD)
		8325 / 925 = 9 

848844 / 804 = 11
184875 / 10875 = 17
1484848 / 14008 = 106 (JD)
484847 / 4007 = 121 (JD)
		7843 / 713 = 11 (JD)
8466 / 166 = 51
84296875 / 1296875 = 65 (JD)
840375 / 10375 = 81 (JD)
84 / 1 = 84
18841784 / 181171 = 104 (JD)
		6842 / 622 = 11 (JD)
384 / 32 = 12 (JD)
845625 / 25625 = 33
8405 / 205 = 41 (JD)
84 / 2 = 42 
		5841 / 531 = 11 (JD)
84375 / 3375 = 25
8424 / 324 = 26
84 / 3 = 28
784845 / 7335 = 107 (JD)
		484 / 44 = 11 (JD)
84 / 4 = 21
3684884 / 36484 = 101 (JD)
		2849 / 259 = 11 (JD)
1848 / 168 = 11 (JD)
845 / 65 = 13
84 / 6 = 14
18848418 / 186618 = 101 (JD)
4845984 / 46596 = 104 (JD)
		847 / 77 = 11
84 / 7 = 12
7848407 / 77707 = 101 (JD)
		2841318466 / 28131866 = 101 (JD)
84375 / 9375 = 9 

858855 / 805 = 11
1859375 / 109375 = 17 (JD)
		7854 / 714 = 11 (JD)
1856 / 116 = 16 (JD)
8575 / 175 = 49
8568 / 168 = 51
855 / 15 = 57
854 / 14 = 61
853125 / 13125 = 65
852 / 12 = 71
8512 / 112 = 76
8505 / 105 = 81 (JD)
85 / 1 = 85 
		6853 / 623 = 11 (JD)
8559375 / 259375 = 33 (JD)
85075 / 2075 = 41 (JD)
		5852 / 532 = 11 (JD)
8528 / 328 = 26
8559375 / 259375 = 33 
		4851 / 441 = 11 (JD)
18525 / 1425 = 13 (JD)
855 / 45 = 19
8505 / 405 = 21 (JD)
		385 / 35 = 11 (JD)
85 / 5 = 17
3085853 / 30553 = 101 (JD)
		1859 / 169 = 11 (JD)
19858519 / 196619 = 101 (JD)
		858 / 78 = 11
8858508 / 87708 = 101 (JD)
18585 / 177 = 105 (JD)
		3851318567 / 38131867 = 101 (JD)
855 / 95 = 9 

868866 / 806 = 11
18668672 / 106072 = 176 (JD)
		7865 / 715 = 11 (JD)
867 / 17 = 51
86328125 / 1328125 = 65 (JD)
8625 / 125 = 69
860625 / 10625 = 81 (JD)
86 / 1 = 86 
		6864 / 624 = 11 (JD)
286 / 22 = 13 (JD)
868 / 28 = 31
86625 / 2625 = 33
864 / 24 = 36
861 / 21 = 41
86 / 2 = 43 
		5863 / 533 = 11 (JD)
8632 / 332 = 26 
		4862 / 442 = 11 (JD)
861 / 41 = 21
8698624 / 49424 = 176 (JD)
		3861 / 351 = 11 (JD)
864 / 54 = 16
860625 / 50625 = 17 (JD)
869868 / 5958 = 146 (JD)
		286 / 26 = 11 (JD)
2086862 / 20662 = 101 (JD)
		869 / 79 = 11
9868609 / 97709 = 101 (JD)
		86965517241386 / 896551724138 = 97 (JD)
4861318668 / 48131868 = 101 (JD)
		86625 / 9625 = 9 

878877 / 807 = 11 
7876 / 716 = 11 (JD)
187 / 11 = 17 (JD)
8772 / 172 = 51
8734375 / 134375 = 65
87075 / 1075 = 81 (JD)
87 / 1 = 87 
		6875 / 625 = 11 (JD)
1875 / 125 = 15 (JD)
8765625 / 265625 = 33
875 / 25 = 35
87125 / 2125 = 41 
		5874 / 534 = 11 (JD)
875 / 35 = 25
8736 / 336 = 26
87 / 3 = 29 
		4873 / 443 = 11 (JD)
8715 / 415 = 21 
		3872 / 352 = 11 (JD)
87125 / 5125 = 17 
		2871 / 261 = 11 (JD)
8775 / 675 = 13 
		187 / 17 = 11 (JD)
1087871 / 10771 = 101 (JD)
		8768719 / 86819 = 101 (JD)
8775 / 975 = 9
87397587 / 939759 = 93 (JD)
38787875 / 399875 = 97 

888888 / 808 = 11
888888 / 8008 = 111 (JD)
		7887 / 717 = 11 (JD)
1888 / 118 = 16 (JD)
1880625 / 110625 = 17 (JD)
888125 / 18125 = 49
8874 / 174 = 51
885 / 15 = 59
8845 / 145 = 61
88359375 / 1359375 = 65 (JD)
8816 / 116 = 76
880875 / 10875 = 81 (JD)
88 / 1 = 88 
		6886 / 626 = 11 (JD)
886875 / 26875 = 33
8815 / 215 = 41
88 / 2 = 44
13888875 / 132275 = 105 (JD)
		5885 / 535 = 11 (JD)
884 / 34 = 26 
		4884 / 444 = 11 (JD)
288 / 24 = 12 (JD)
1885 / 145 = 13 (JD)
882 / 42 = 21
88 / 4 = 22
228888 / 2244 = 102 (JD)
888888 / 4884 = 182 (JD)
		3883 / 353 = 11 (JD)
881875 / 51875 = 17 
		2882 / 262 = 11 (JD)
1881 / 171 = 11 (JD)
11888811 / 117711 = 101 (JD)
		88 / 8 = 11
8888 / 88 = 101 (JD)
		88875 / 9875 = 9
8888875 / 99875 = 89 

898899 / 809 = 11 
7898 / 718 = 11 (JD)
189125 / 11125 = 17 (JD)
8976 / 176 = 51
896 / 16 = 56
89375 / 1375 = 65
891 / 11 = 81
89 / 1 = 89 
		6897 / 627 = 11 (JD)
28925 / 2225 = 13 (JD)
899 / 29 = 31
8971875 / 271875 = 33
89175 / 2175 = 41 
		5896 / 536 = 11 (JD)
8944 / 344 = 26
1894899 / 13439 = 141 (JD)
		4895 / 445 = 11 (JD)
8925 / 425 = 21 
		3894 / 354 = 11 (JD)
896 / 56 = 16
8925 / 525 = 17
8998996 / 59596 = 151 (JD)
		2893 / 263 = 11 (JD)
1892 / 172 = 11 (JD)
12898912 / 127712 = 101 (JD)
		891 / 81 = 11
1898901 / 18801 = 101 (JD)
8918925 / 81825 = 109 (JD)
		

90990 / 90 = 11
390 / 30 = 13 (JD)
190 / 10 = 19 (JD)
9090909 / 90009 = 101 (JD)
9909190 / 90910 = 109 (JD)
3907790 / 30770 = 127 (JD)
1901790 / 10170 = 187 (JD)
		7909 / 719 = 11 (JD)
1901875 / 111875 = 17 (JD)
9078 / 178 = 51
90390625 / 1390625 = 65 (JD)
901125 / 11125 = 81
90 / 1 = 90 
		6908 / 628 = 11 (JD)
9075 / 275 = 33
902 / 22 = 41
90 / 2 = 45 
		5907 / 537 = 11 (JD)
90625 / 3625 = 25
9048 / 348 = 26
90 / 3 = 30
190905 / 1335 = 143 (JD)
		4906 / 446 = 11 (JD)
903 / 43 = 21 
		3905 / 355 = 11 (JD)
903125 / 53125 = 17
90 / 5 = 18 
		2904 / 264 = 11 (JD)
90 / 6 = 15
19090 / 166 = 115 (JD)
		1903 / 173 = 11 (JD)
13909013 / 137713 = 101 (JD)
		902 / 82 = 11
2909002 / 28802 = 101 (JD)
		90 / 9 = 10
909909 / 9999 = 91 

919911 / 901 = 11 
891 / 81 = 11 (JD)
19125 / 1125 = 17 (JD)
91875 / 1875 = 49
918 / 18 = 51
915 / 15 = 61
9140625 / 140625 = 65
9125 / 125 = 73
912 / 12 = 76
91125 / 1125 = 81
91 / 1 = 91
8091918 / 80118 = 101 (JD)
3919104 / 31104 = 126 (JD)
		6919 / 629 = 11 (JD)
9178125 / 278125 = 33
91225 / 2225 = 41 
		5918 / 538 = 11 (JD)
9152 / 352 = 26 
		4917 / 447 = 11 (JD)
19175 / 1475 = 13 (JD)
9135 / 435 = 21 
		3916 / 356 = 11 (JD)
91375 / 5375 = 17
9169916 / 56956 = 161 (JD)
		2915 / 265 = 11 (JD)
1914 / 174 = 11 (JD)
91 / 7 = 13
14919114 / 147714 = 101 (JD)
6910991 / 67097 = 103 (JD)
9156917 / 75677 = 121 (JD)
		913 / 83 = 11
3919103 / 38803 = 101 (JD)
		59177528091 / 597752809 = 99 (JD)
910109182 / 9010982 = 101 (JD)

929922 / 902 = 11
1924925 / 10405 = 185 (JD)
		8921 / 811 = 11 (JD)
492 / 41 = 12 (JD)
1923125 / 113125 = 17 (JD)
9282 / 182 = 51
9275 / 175 = 53
92625 / 1625 = 57
92421875 / 1421875 = 65 (JD)
924 / 14 = 66
923 / 13 = 71
921375 / 11375 = 81
92 / 1 = 92 
		792 / 72 = 11 (JD)
2925 / 225 = 13 (JD)
192 / 12 = 16 (JD)
928125 / 28125 = 33
925 / 25 = 37
9225 / 225 = 41
92 / 2 = 46
7092927 / 70227 = 101 (JD)
292392 / 2232 = 131 (JD)
		5929 / 539 = 11 (JD)
9256 / 356 = 26 
		4928 / 448 = 11 (JD)
924 / 44 = 21
92 / 4 = 23 
		3927 / 357 = 11 (JD)
928 / 58 = 16
924375 / 54375 = 17 
		2926 / 266 = 11 (JD)
192 / 16 = 12 (JD)
1192992 / 11696 = 102 (JD)
		1925 / 175 = 11 (JD)
15929215 / 157715 = 101 (JD)
		924 / 84 = 11
4929204 / 48804 = 101 (JD)
92925 / 885 = 105 (JD)
		1920109283 / 19010983 = 101 (JD)

939933 / 903 = 11
39325 / 3025 = 13 
		8932 / 812 = 11 (JD)
193375 / 11375 = 17 (JD)
9384 / 184 = 51
934375 / 14375 = 65
9315 / 115 = 81
93 / 1 = 93 
		7931 / 721 = 11 (JD)
9384375 / 284375 = 33
936 / 26 = 36
93275 / 2275 = 41 
		693 / 63 = 11 (JD)
9375 / 375 = 25
936 / 36 = 26
93 / 3 = 31
6093936 / 60336 = 101 (JD)
2939328 / 23328 = 126 (JD)
1939375 / 13375 = 145 (JD)
		4939 / 449 = 11 (JD)
9345 / 445 = 21 
		3938 / 358 = 11 (JD)
935 / 55 = 17 
		2937 / 267 = 11 (JD)
1936 / 176 = 11 (JD)
16939316 / 167716 = 101 (JD)
		935 / 85 = 11
5939305 / 58805 = 101 (JD)
		193939375 / 1999375 = 97 (JD)
2930109384 / 29010984 = 101 (JD)

949944 / 904 = 11 
8943 / 813 = 11 (JD)
294 / 21 = 14 (JD)
1944375 / 114375 = 17 (JD)
949375 / 19375 = 49
9486 / 186 = 51
9455 / 155 = 61
945 / 15 = 63
94453125 / 1453125 = 65 (JD)
9424 / 124 = 76
941625 / 11625 = 81
94 / 1 = 94 
		7942 / 722 = 11 (JD)
94875 / 2875 = 33
943 / 23 = 41
94 / 2 = 47 
		6941 / 631 = 11 (JD)
9464 / 364 = 26
945 / 35 = 27
9425 / 325 = 29 
		594 / 54 = 11 (JD)
945 / 45 = 21
5094945 / 50445 = 101 (JD)
		3949 / 359 = 11 (JD)
945625 / 55625 = 17 
		2948 / 268 = 11 (JD)
1947 / 177 = 11 (JD)
9425 / 725 = 13
17949417 / 177717 = 101 (JD)
94794 / 777 = 122 (JD)
		946 / 86 = 11
6949406 / 68806 = 101 (JD)
		3940109485 / 39010985 = 101 (JD)

959955 / 905 = 11 
8954 / 814 = 11 (JD)
1955 / 115 = 17 (JD)
9588 / 188 = 51
9546875 / 146875 = 65
95175 / 1175 = 81
95 / 1 = 95 
		7953 / 723 = 11 (JD)
29575 / 2275 = 13 (JD)
1952 / 122 = 16 (JD)
9590625 / 290625 = 33
95325 / 2325 = 41
1295956 / 12226 = 106 (JD)
		6952 / 632 = 11 (JD)
195 / 13 = 15 (JD)
9568 / 368 = 26 
		5951 / 541 = 11 (JD)
9555 / 455 = 21 
		495 / 45 = 11 (JD)
195 / 15 = 13 (JD)
95625 / 5625 = 17
95 / 5 = 19
4095954 / 40554 = 101 (JD)
49595 / 455 = 109 (JD)
1959552 / 15552 = 126 (JD)
1953895 / 15385 = 127 (JD)
9552195 / 55215 = 173 (JD)
950895 / 5085 = 187 (JD)
		2959 / 269 = 11 (JD)
1958 / 178 = 11 (JD)
18959518 / 187718 = 101 (JD)
		957 / 87 = 11
7959507 / 78807 = 101 (JD)
		4950109586 / 49010986 = 101 (JD)
950526315795 / 9052631579 = 105 (JD)

969966 / 906 = 11
3965 / 305 = 13 
		8965 / 815 = 11 (JD)
1965625 / 115625 = 17 (JD)
969 / 19 = 51
96484375 / 1484375 = 65 (JD)
9625 / 125 = 77
961875 / 11875 = 81
96 / 1 = 96 
		7964 / 724 = 11 (JD)
969375 / 29375 = 33
9635 / 235 = 41
96 / 2 = 48 
		6963 / 633 = 11 (JD)
396 / 33 = 12 (JD)
96875 / 3875 = 25
9672 / 372 = 26
961 / 31 = 31
96 / 3 = 32
59696 / 533 = 112 (JD)
		5962 / 542 = 11 (JD)
196 / 14 = 14 (JD)
966 / 46 = 21
96 / 4 = 24 
		4961 / 451 = 11 (JD)
966875 / 56875 = 17 
		396 / 36 = 11 (JD)
96 / 6 = 16
3096963 / 30663 = 101 (JD)
19621965 / 162165 = 121 (JD)
9696875 / 66875 = 145 (JD)
		1969 / 179 = 11 (JD)
19969619 / 197719 = 101 (JD)
		968 / 88 = 11
96 / 8 = 12
8969608 / 88808 = 101 (JD)
596496 / 5848 = 102 (JD)
		969696875 / 9996875 = 97 (JD)
5960109687 / 59010987 = 101 (JD)

979977 / 907 = 11 
8976 / 816 = 11 (JD)
197625 / 11625 = 17 (JD)
9792 / 192 = 51
976 / 16 = 61
975 / 15 = 65
9728 / 128 = 76
972 / 12 = 81
97 / 1 = 97
4979755 / 41155 = 121 (JD)
		7975 / 725 = 11 (JD)
9796875 / 296875 = 33
975 / 25 = 39
97375 / 2375 = 41 
		6974 / 634 = 11 (JD)
9776 / 376 = 26 
		5973 / 543 = 11 (JD)
9765 / 465 = 21 
		4972 / 452 = 11 (JD)
9775 / 575 = 17 
		3971 / 361 = 11 (JD)
975 / 65 = 15 
		297 / 27 = 11 (JD)
975 / 75 = 13
2097972 / 20772 = 101 (JD)
2972975 / 27275 = 109 (JD)
979776 / 7776 = 126 (JD)
		979 / 89 = 11
9979709 / 98809 = 101 (JD)
		6970109788 / 69010988 = 101 (JD)
972631578975 / 9263157895 = 105 (JD)

989988 / 908 = 11 
8987 / 817 = 11 (JD)
1986875 / 116875 = 17 (JD)
198 / 11 = 18 (JD)
9894 / 194 = 51
98515625 / 1515625 = 65 (JD)
982125 / 12125 = 81
98 / 1 = 98 
		7986 / 726 = 11 (JD)
1984 / 124 = 16 (JD)
984 / 24 = 41
98 / 2 = 49 
		6985 / 635 = 11 (JD)
988 / 38 = 26 
		5984 / 544 = 11 (JD)
987 / 47 = 21 
		4983 / 453 = 11 (JD)
19825 / 1525 = 13 (JD)
988125 / 58125 = 17 
		3982 / 362 = 11 (JD)
2981 / 271 = 11 (JD)
98 / 7 = 14 
		198 / 18 = 11 (JD)
1098981 / 10881 = 101 (JD)
		7980109889 / 79010989 = 101 (JD)

999999 / 909 = 11
39975 / 3075 = 13
1995 / 105 = 19
999999 / 9009 = 111 (JD)
3999975 / 30075 = 133 (JD)
		8998 / 818 = 11 (JD)
19975 / 1175 = 17 (JD)
9996 / 196 = 51
9975 / 175 = 57
9953125 / 153125 = 65
994 / 14 = 71
99225 / 1225 = 81
99 / 1 = 99 
		7997 / 727 = 11 (JD)
99425 / 2425 = 41 
		6996 / 636 = 11 (JD)
299 / 23 = 13 (JD)
9984 / 384 = 26
992 / 32 = 31
99 / 3 = 33
229999 / 2233 = 103 (JD)
11991992 / 113132 = 106 (JD)
		5995 / 545 = 11 (JD)
9975 / 475 = 21 
		4994 / 454 = 11 (JD)
99875 / 5875 = 17 
		3993 / 363 = 11 (JD)
992 / 62 = 16
5995996 / 56566 = 106 (JD)
999922 / 6622 = 151 (JD)
		2992 / 272 = 11 (JD)
19969925 / 176725 = 113 (JD)
		1991 / 181 = 11 (JD)
11999911 / 118811 = 101 (JD)
		99 / 9 = 11
9999 / 99 = 101 (JD)


Problem #3:

We give solutions where the sequence has length at least k+3. Marc Lapierre found many that I missed.

n \ k2345
13 55575777272228288868666 (ML)
14 6588266 55454446466626222 (ML)
1768517 99962227888 9939933133116116626622 (ML)
1919576 722855133 77714446555 31331939938988958558 (ML)
21 7677679779737737 (ML)
2346923 5759897 44393381886 5554533373 (ML)
24 552888
26 77272228288868666 (ML)
28 336448
29 114848836366 (ML)
31 1558992
32 544800
34 911224488 16116626622 (ML)
35 7735005
37 48843663 3363344844 (ML)
38 772922268666 (ML)
43 2924515
46 8832668
47 48883666
51 11224488
52 2111284448 (ML)
53 969964664 449449979977 (ML)
54 66636666 (ML)
56 333768880 (ML)
57 833345999 (ML)
61 993696624644 (ML)
62 99944888 (ML)
63 77742777 (ML)
64 8448000
66 77748444 (ML)
67 88442211 22244555 (ML)
68 1224488
69 44436777 (ML)
71 99968444 (ML)
72 889928800 (ML)
73 77745333 (ML)
74 363344844 (ML)
77 33341888 (ML)
78 212168484 (ML)
79 55537999 (ML)
81 99954999 (ML)
83 55527888 (ML)
88 33352888 (ML)
93 44454777 (ML)
95 88825555 (ML)
98 55566444 (ML)
102 11224488
104 22256888 (ML)
106 69964664
107 77789666 (ML)
109 77717222 (ML)
112 77728000 (ML)
116 11136888
121 99946222
122 93696624644
124 99944888
128 884480000
132 77748444
133 88844555
136 77792888
142 99968444
144 99936000
148 63344844 (ML)
156 212168484
164 48872488 (ML)
169 77740222
183 99918666
192 66624000
199 22288555
201 88442211
204 1224488
224 77728000
229 3664122
232 11136888
238 66640222
242 99946222
244 66612444
248 99944888
249 55527888
256 88832000
277 3213296396
283 933962264
284 99968444
288 99936000
338 77740222
357 99960333
361 3321299636
366 99918666
377 8444863336
453 448471121
566 33962264
640 84480000
1061 84880636
1107 99630996
1923 21153844
2459 6639344262
4878 68292682
4918 639344262

Problem #4:

Here are the smallest-known solutions. Joe DeVincentis and Alex Rower proved some cases impossible.

d(z)=2
d(y) \ d(x)23456789101112
22+3=5
33+4=74+9=13
43+8=119+8=1715+8=23
53+16=1925+16=4115+16=3181+16=97
65+12=179+20=2921+20=4116+45=6145+28=73
73+64=679+64=7315+64=7916+15625=1564145+64=109none (JD)
85+24=2925+42=6735+24=5981+56=13745+56=10164+135=199135+56=191
95+36=4125+36=6135+36=7181+100=18163+100=163729+100=82956+225=281225+196=421
105+48=5325+48=7335+48=8381+112=19345+112=15764+567=631189+80=269225+112=337405+208=613
117+1024=10319+1024=103315+1024=1039819628286980801 + 16 = 819628286980817 (JD)45+1024=1069729+1024=1753105+1024=1129225+1024=1249405+1024=1429none (JD)
127+60=6725+72=9735+72=10781+160=24163+160=22364+315=379189+160=349225+224=449176+315=4911024+525=1549495+224=719

d(z)=3
d(y) \ d(x)23456789101112
22+2=4
35+4=9none (AR)
43+6=94+21=2515+10=25
5none (JD)9+16=2533+16=49none (JD)
65+20=254+45=4921+28=4916+153=16976+45=121
7none (JD)none (JD)57+64=121none (JD)none (JD)none (JD)
87+42=499+40=4991+30=12116+105=12199+70=16964+105=169184+105=289
913+36=49none (JD)21+100=121none (JD)325+36=36164+225=289189+100=289none (JD)
1041+80=1219+112=12157+112=16981+208=289153+208=361729+112=841665+176=841225+304=5291377+304=1681
11none (JD)none (JD)none (JD)none (JD)657+1024=1681none (JD)345+1024=1369none (JD)none (JD)none (JD)
1213+108=12125+96=12185+84=16916+825=841369+160=52964+1305=1369165+364=529256+585=841405+1276=16811024+825=1849825+544=1369

d(z)=4
d(y) \ d(x)23456789101112
23+3=6
32+4=64+4=8
42+6=84+6=108+6=14
55+16=2149+16=656+16=22625+81=706
62+12=149+12=2110+12=2216+18=3420+18=38
75+64=69121+64=18510+64=7481+64=14518+64=82729+64=793
82+24=269+24=3310+24=3416+30=4618+40=5864+30=9442+40=82
92+36=3849+36=8510+36=4681+484=56550+36=8664+441=50570+36=106256+225=481
103+48=519+48=5710+48=5881+48=12945+48=9364+162=22670+48=118100+162=262162+112=274
113+1024=102749+1024=107322+1024=1046625+1024=164918+1024=104215625+59049=74674 (JD)70+1024=1094441+1024=1465162+1024=11861024+59049=60073 (JD)
122+60=629+60=6914+60=7416+90=10628+90=11864+150=21470+72=142100+126=226112+90=2021024+90=1114224+90=314

d(z)=5
d(y) \ d(x)23456789101112
213+3=16
37+9=16none (AR)
42+14=164+77=8110+6=16
5none (JD)none (JD)65+16=81none (JD)
65+76=814+12=166+75=81none (JD)63+18=81
717+64=81none (JD)none (JD)none (JD)none (JD)none (JD)
83+78=8125+56=8115+66=8116+609=625153+472=62564+561=625595+30=625
9none (JD)none (JD)589+36=625none (JD)45+36=81none (JD)184+441=625none (JD)
10577+48=625none (JD)33+48=81none (JD)52+2349=2401none (JD)513+112=625none (JD)13264+1377=14641
11none (JD)none (JD)none (JD)none (JD)none (JD)none (JD)27537+1024=28561none (JD)1377+1024=2401none (JD)
125+620=6259+72=8121+60=8181+544=62545+580=62564+130257=13032140+585=625100+525=625405+220=6251024+13617=146412275+126=2401

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 7/31/19.