Direct Proofs (3)

Write clear and complete direct proofs for each of the following mathematical statements.

If m is even and n is odd, then mn + 3n is odd.

Proof. Assume that m is even and n is odd. So m = 2k and n = 2j + 1 for some integers k and j. Then
           mn + 3n = (2k)(2j + 1) + 3(2j + 1)
                         = 4kj + 2k + 6j + 3
                         = 4kj + 2k + 6j + 2 + 1
                         = 2(2kj + k + 3j + 1) + 1
                         = 2q + 1

where q = 2kj + k + 3j + 1 is an integer. So mn + 3n is odd.

QED