Find all possible two digit positive integers N for which the sum of digits of 10^N - N is divisible by 170.

The sum of the digits of 10^N-N is 9*(N-2) plus the sum of the digits of 100-N.  It is easy to write a program that computes this number for each N in the closed interval [10,99] and checks whether or not it is divisble by 170. 5 of those numbers are divisible by 170, namely 170, 340, 510, 680, and 850 corresponding repectively to N=20, 39, 58, 77, and 96 resp. Notice that these N's are in an arithmetic progression with common difference of 19.

It remains to be seen what completely analytic solution is possible for this problem.

Posted by Richard on 2006-07-14 13:10:55