Determine truth value: If $n^2$ is a multiple of 5, then $n$ is a multiple of 5.

Question

good day mathematicians of Math Stack Exchange, I have a bit of curiosity about this exercise that my teacher proposed in class today and says the following:

Determine truth value:

For all $n\in\mathbb{Z}$, if $n^2$ is a multiple of 5, then n is a multiple of 5.

Thanks so much! Please give a proof or a counterexample.

Quote of the day:

"If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it."

George Pólya

1887-1985

Answer 1

Let $y$ be any integer. Express $|y|=5q+r$. Where $r$ is the remainder of $|y|$ after division by $5$. And $q$ is the quotient.

Then,

$$y^2=|y|^2=(5q+r)^2=5q^2+10qr+r^2=5(q^2+2qr)+r^2$$

$$=5k+r^2$$

If we want $y^2$ to be divisible by $5$ then $y^2-5k=r^2$ needs to be divisible by $5$.

There are a couple possibilities for the remainder $r$, those being $0$, $1$, $2$, $3$, and $4$. Only $r=0$ gives an $r^2$ divisible by $5$. So we need $|y|=5q+0$ and hence $y=\pm 5q$.