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 $\neg B \implies \neg A$ 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.