Definition: An integer $x$ is called a perfect square if there is an integer $y$ such that $x=y^2.$ For example, $0,1,4,$ and $9$ are perfect squares while $2,3,5,$ and $6$ are not.

The following proof shows that if $m$ and $n$ are perfect squares, then $mn($the product of $m$ and $n)$ is also a prefect square.

Let $m$ and $n$ be perfect squares.

By the definition above, there must be integers $s$ and $t$ such that $m=s^2$ and $n=t^2.$

Then, we can write $mn=s^2{t}^2=(st{)}^2.$

If $s$ and $t$ are integers, so is st$.$

Rename $st=a.$

Then, $mn=(st{)}^2=a^2$ for an integer $a.$

Again, by the definition above, this means that $mn$ is a perfect square.

Therefore, if $m$ and $n$ are perfect squares, then $mn$ is also a perfect square. Q$.$E$.$D$.$

Which of the following is true about this proof?

It is invalid because there is a mistake in the proof.

It is valid and it uses proof by contraposition.

It is valid LAd it is a direct proof.

It is valid and it uses proof by contradiction.