No subject

[David Wilson]

For n coprime to 10, empirical evidence suggests that any number of n digits or 
more has a substring divisible by n.  Supposing this to be true, and supposing 
my program is correct, the following are the largest numbers a(n) with no 
substring divisible by n for 3 <= n <= 63, gcd(n, 10) = 1.

n: a(n)

3: 88
7: 999993
9: 88888888
11: 9898654321
13: 999998986863
17: 9999999999924214
19: 999999999999962944
21: 99999899999899978513
23: 9999999999999999976315
27: 99899899899899899899899899
29: 9999999999999999999999912715
31: 999999999999997999989532754469
33: 98989898989898988967979898989897
37: 998998998998998998998998913891442761
39: 99999899999899999587989698968431197217
41: 9999899998989989989788887889988432561871
43: 999999999999999999998888897889788896248422
47: 9999999999999999999999999999999999999999242736
49: 999999999999999999999999999999999999999377562877
51: 99999999999999989999999999999998999999981333996729
53: 9999999999998998888869998889899869856598491195266386
57: 99999999999999999899999999999999999899999999366993663214
59: 9999999999999999999999999999999999999999999996456465846819
61: 999999999999999999999999999999999999999999998582455227165727
63: 99999899999899999899999899999899999899999899999899966241755974

Some questions:

- Is it indeed true that if gcd(n, b) = 1, a base-b n-digit number has a 
substring divisible by n?

- Are the above values of a(n) correct?

- Are there any exploitable patterns in the numbers above that might be used to 
find them more quickly/elegantly (than the brute-force bounded depth-first 
search I used)?

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- David Wilson