1. The problem statement, all variables and given/known data Prove that for all integers x >= 8, x can be written in the form 3m + 5n, where m and n are non-negative integers. 2. Relevant equations 3. The attempt at a solution Proof by induction on n that every integer n >= 8 can be expressed as n = 5x + 3y, with some integers x and y. Let n = 8. Then n = 8 = 5(1) + 3(1), so the proposition is true for the base case. Suppose the proposition is true for some number integer n = k > 8, i.e. k = 5x + 3y, for integers x and y. Consider the case when n = k + 1. Then we have k + 1 = 5x + 3y + 1 = 5x + 3y + 1 + 5 - 5 = 5x - 5 + 3y + 6 = 5(x - 1) + 3(y + 2). Since the proposition is true for the base case and it being true for n = k implies that it is true for n = k + 1, then n = 5x + 3y for some integers x and y. I think that's almost it, but what about showing that m and n will never have to be negative?