Use an indirect proof: If n^2 + 31 is odd, then n must be even.

Answer 1

Suppose not...

suppose n^2 + 31 is odd while n is odd...

then n = 2t+1 for some integer t

n^2 + 31 = (2t+1)^2 + 31 = 4t^2 + 4t + 32 = 4(t^2 + t+ 32)

which is clearly even, a contradiction