Let n ∈ Z. Prove if n − 2 is even, then 3n 2 + 5 is odd.

Answer 1

If n-2 is even, n is even. To show: if n - 2 is even, n - 2 = 2x, an integer. Then, n = n-2 + 2 = 2x + 2 = 2(x+1) is even.

Then, 3n² + 5 = 3 * n * n + 5 = odd * even * even + odd = (odd * even) * even + odd = even * even + odd = even + odd = odd