N ends with 56, so
N=100*m+56
N is divisible by 56, and lcm(56,100) = 1400, therefore
N=1400*k+56
The sum of N's digits is 56, so the sum of digits of 14*k is 45.
99999 ≠ 14*k , so we need at least six digits. Let the six digits be abcdef.
We want
100000a + 10000b + 1000c + 100d + 10e + f = 0 (mod 14), with a+b+c+d+e+f = 45
Substituting a' = 9a, b' = 9b etc.:
a'+b'+c'+d'+e'+f' = 9
7 = 2a' + 4b' + 6c' + 2d'  4e' + f' (mod 14)
Skipping a few steps, the best solution is abcdef = 298998
N = 29899856

Posted by Tristan
on 20131005 01:27:53 