Beginner PROOF question involving perfect squares.

Question

Let m, n be two positive integers. Prove that if m, n are perfect sqaures, then the product mn is also a perfect square.

So I'm not really sure where to start on that example. I assume that we can use direct proof here. I'm not sure how to write up perfect squares in formula.

Solutions?

Answer 1

If $m$ is a perfect square, this means that there is a positive integer $k$ such that $m=k^2$. Similarly for $n$. So what can we say about $mn$?

Answer 2

Hint $m$ is a perfect square means that there exists an integer $a$ such that $$m=a^2$$ Same way, there exists an integer $b$ so that $n=b^2$.

Answer 3

The definition of a perfect square $a$ is $$a=b^2,$$

where $b$ is an integer.

So: $$m=k^2, n=t^2,$$ their product: $$mn=k^2t^2=(kt)^2$$

and since the product of two integers is another integer, we can let $kt=r$ and $l=r^2$.

$$mn=k^2t^2=(kt)^2=r^2=l.$$

$l$ fits the definition of a perfect square.