Question

Prove:

For all integers n, if n2 is odd, then n is odd.
Use a proof by contraposition, as in Lemma 1.1.
Let n be an integer. Suppose that n is even, i.e., n = for some integer k. Then n2 = = 2



is also even.

Answer 1

Given the statement, "If
`n^2` is odd, then
`n` is odd," its contrapositive claims that, "If
`n` is not odd, then
`n^2` is not odd."

So assume
`n` is not odd, i.e.
`n` is even. This means there is an integer
`k` for which
`n=2k`. Squaring this gives
`n^2=(2k)^2=4k^2`.

Well, we can write
`4k^2=2(2k^2)`, and
`2k^2` is just another integer, which means
`4k^2=(2k)^2=n^2` must be even.