Simple Proof Methods

Another example:

Theorem. If $n^2$ is divisible by $4$, then $n$ is even.

Proof. We will prove the contrapositive of this statement, i.e., "if $n$ is odd, then $n^2$ is not divisible by $4$".

Let $n$ be indeed an odd integer. By the definition of odd integers, we have $n = 2k + 1$ for some $k \in \mathbb{Z}$. We have: $$n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1,$$ which isn't divisible by $4$, as we wished to prove, because $4(k^2 + k) + 1$ leaves a remainder of $1$ when divided by $4$.

Hence, the statement "if $n^2$ is divisible by $4$, then $n$ is even" is true.◼