2010/035) Show that

27 x (23^n) + 17 x (10^(2n)) is divisible by 11 for all positive integers n

proof:
23 = 1 mod 11
so 23^n = 1 mod 11

10^2 = 100 = 1 mod 11

so 10^2n =1 mod 11

so 27 x (23^n) + 17 x (10^(2n)) = 27 * 1 + 17 * 1 = 44 = 0 mod 11

so divisible by 11