Question

Prove by contrapositive that for all integers n, if n2 is odd, the n is odd.

Answer 1

Proving a statement by contrapositive means that if you want to prove that
`A \implies B`, you can instead prove that
`\\eg B \implies \\eg A` since the two are equivalent.

So, proving that
`n^2 \text{ is odd} \implies n \text{ is odd}` by contrapositive means to prove that

`n \text{ is even} \implies n^2 \text{ is even}`

since of course the negation of being odd is being even.

This claim is quite easy to prove: if n is even, then it is twice some other integer k:
`n=2k`

This means that its square is
`n^2 = 4k^2 = 2(2k^2)`

And so, if n is even, its square is also even. This proves, by contrapositive, that if n squared is odd, then n is odd.