Sum of digits - experience #23

For a positive integer k, define d(k) as the sum of the digits of k in its usual decimal representation. Thus d(42) = 4+2 = 6.

For a positive integer n, define S(n) as the number of positive integers k < 10ⁿ with the following properties :

• k is divisible by 23 and
• d(k) = 23.

Find S(11¹²) and give your answer mod 10⁹.

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter