Let f(x) = x^2 - 12x + 20 and S = n Z : f(n) k , where k is an integer. If _ n S f(n) = 900 , then the maximum possible value of k is _______.

Step-by-step solution

Given f(x) = x^2 - 12x + 20 , we can rewrite it by completing the square as f(x) = (x-6)^2 - 16 . The condition for the set S is f(n) k , which means (n-6)^2 - 16 k , or (n-6)^2 k+16 . Let m = n-6 . As n ranges over integers, m ranges over integers. The condition becomes m^2 k+16 . Let M be the largest integer such that M^2 k+16 . Then m takes all integer values from -M to M . The sum is given by: _ n S f(n) = _ m=-M ^ M (m^2 - 16) = 900 We can split this sum as: _ m=-M ^ M m^2 - _ m=-M ^ M 16 = 2 _ m=1 ^ M m^2 - 1