Let the 10 digits of the number be a-j, eg:
abcdefghij
abcdefghi0 div by 10
abcd5fghi0 div by 5
div by 2
even digits must be in 2, 4, 6, 8, 0
odd digits must be in 1, 3, 5, 7, 9.
div by 3
a+b+c mod 3 = 0
d+e+f mod 3 = 0
g+h+i mod 3 = 0
div by 4
Since c is odd, d must be 2 or 6.
similarly, since g is odd, h must be 2 or 6.
-> b and f are 4 and 8 in some order.
div by 8
Since f is 4 or 8, gh mod 8 = 0. Possibilities for gh:
32, 72, 16, 96
combined with div by 9 (3), the possibilities for ghi:
321, 327, 723, 729, 963,
d + 5 + f (mod 3) = 0
d + f = 1 (mod 3)
possibilities for def, given that f = 4 or 8 and e = 5
654, 258
do div by 7 seperately. So combining the options, and then taking mod of the first 7 digits we have:
987654321 => 5 mod 7
789654321 => 4 mod 7
981654327 => 2 mod 7
189654327 => 5 mod 7
981654723 => 6 mod 7
189654723 => 2 mod 7
183654729 => 6 mod 7
381654729 => 0 mod 7
147258963 => 6 mod 7
741258963 => 2 mod 7
Answer is
3816547290
8,127
