Prove that for all integers m and n, m + n and m - n are either both odd or both even.?

take numbers mod 2
even ≡ 0 mod 2
odd ≡ 1 mod 2

m ≡ p mod 2
n ≡ q mod 2

m+m ≡ p+q mod 2
m-n ≡ p-q mod 2

if p≡q then p-q is 0
p+q = 0 or 2. Both are ≡ 0 mod 2.

if p not ≡ q mod 2 then p+q ≡ 1 mod 2
p-q ≡ 1 or -1. Both are ≡ 1 mod 2.

Therefore both are even or both are odd.
QED