While others have already mentioned the divisibility of decimal repunits,
$$m := \frac{{10}^n - 1}{9}.$$
1 1
2 11 11
3 111 3 37
4 1111 11 101
5 11111 41 271
6 111111 3 7 11 13 37
7 1111111 239 4649
8 11111111 11 73 101 137
9 111111111 3^2 37 333667
10 1111111111 11 41 271 9091
11 11111111111 21649 513239
12 111111111111 3 7 11 13 37 101 9901
13 1111111111111 53 79 265371653
14 11111111111111 11 239 4649 909091
15 111111111111111 3 31 37 41 271 2906161
16 1111111111111111 11 17 73 101 137 5882353
17 11111111111111111 2071723 5363222357
18 111111111111111111 3^2 7 11 13 19 37 52579 333667
19 1111111111111111111 1111111111111111111
20 11111111111111111111 11 41 101 271 3541 9091 27961
21 111111111111111111111 3 37 43 239 1933 4649 10838689
22 1111111111111111111111 11^2 23 4093 8779 21649 513239
23 11111111111111111111111 11111111111111111111111
24 111111111111111111111111 3 7 11 13 37 73 101 137 9901 99990001
25 1111111111111111111111111 41 271 21401 25601 182521213001
26 11111111111111111111111111 11 53 79 859 265371653 1058313049
27 111111111111111111111111111 3^3 37 757 333667 440334654777631
28 1111111111111111111111111111 11 29 101 239 281 4649 909091 121499449
29 11111111111111111111111111111 3191 16763 43037 62003 77843839397
30 111111111111111111111111111111 3 7 11 13 31 37 41 211 241 271 2161 9091 2906161
31 1111111111111111111111111111111 2791 6943319 57336415063790604359
I think a practical use of this divisibility is frequently overlooked :
- For really gigantic inputs, one may literally stack wide band of digits on top of each others, and sum up the stacks, before performing any sort of modular arithmetic.
e.g. 12 111111111111 : 3 7 11 13 37 101 9901
One can stack every 12-digits directly on top of each other if one wants to calculate any of the combination of all these prime factors, which would massively speed up primality checking for small factors
for instance, to perform a quick test on
7407457283755447477426709403332725993034294823928945951559129
6149310316146786637645823164508144688817069018866259431952169
74
+ 074572837554 = 74572837628
+ 474774267094 = 549347104722
+ 033327259930 = 582674364652
+ 342948239289 = 925622603941
+ 459515591296 = 1385138195237
+ 149310316146 = 1534448511383
+ 786637645823 = 2321086157206
+ 164508144688 = 2485594301894
+ 817069018866 = 3302663320760
+ 259431952169 = 3562095272929
------------------------------
3,562,095,272,929
Let's test them all except 9901 :
3 * 7 * 11 * 13 * 37 * 101 := 11,222,211
( 3562095272929 % 11222211 ) % 3 := ( 8390575 % 3 ) := 1 | 3562095272929 % 3 == 1
( 3562095272929 % 11222211 ) % 7 := ( 8390575 % 7 ) := 4 | 3562095272929 % 7 == 4
( 3562095272929 % 11222211 ) % 11 := ( 8390575 % 11 ) := 6 | 3562095272929 % 11 == 6
( 3562095272929 % 11222211 ) % 13 := ( 8390575 % 13 ) := 11 | 3562095272929 % 13 == 11
( 3562095272929 % 11222211 ) % 37 := ( 8390575 % 37 ) := 11 | 3562095272929 % 37 == 11
( 3562095272929 % 11222211 ) % 101 := ( 8390575 % 101 ) := 0 | 3562095272929 % 101 == 0
Which indeed, has correctly identified 101 as a factor, since the original value was generated via
7407457283755447477426709403332725993034294823928945951559129
6149310316146786637645823164508144688817069018866259431952169: 101^13 23456789^13
So instead of requiring a big-integer library, even before we perform any sort of modular arithmetic on it, this chunking approach already pre-reduced the problem to something that can fit within 64-bit data types.