Chapter 4.2, Problem 18ES

Find the mistakes in the “proofs” show in 15-19.

Theorem: The product of any even integer and n is any odd integer is even.

“Proof: Suppose m is any even integer and n is any odd integer. If $m\cdot n=2r$ . Also since m is even, there exists an integer p such that $m=2p$ , and since n is odd there exists an integer q such that n=2q+1. Thus

$mn=\left(2p\right)\left(2p+1\right)=2r$ .

Where r is an integer. By definition of even, then, m.n is even, as was to be shown.”