Question

Using the definitions of even integer and odd integer, give a proof by contraposition that this statement is true for all integers n : If 3 n - 5 is even, then n is odd.

Answer 1

Suppose $n$ is even, i.e. there is some integer $k$ such that $n=2k$. Then

$3n-5=3(2k)-5=6k-5=6k-6+1=2(3k-3)+1=2K+1$

where $K$ is just some other integer. Therefore $3n-5$ is odd, proving the contrapositive, and so the original statement is also true.