If $P,Q,R$ are natural numbers where $P$ and $Q$ are primes, and $Q$ divides $PR$, which of the following is true?

Question

If $P,Q,R$ are natural numbers where $P$ and $Q$ are primes, and $Q$ divides $PR$, which of the following is true?

1. $P\mid\,Q$
2. $P\mid\,R$
3. $P\mid\,QR$
4. $P\mid\,PQ$

I know that the last option is correct, but how? Thank you in advance.
Edit: The third option is $P|QR$ Original: The third option was $P|PR$.

Answer 1

No matter what the numbers $p$ and $q$ are, $p$ will always divide $pq$ because, by definition:

$x|y$ if and only if there exists some $k\in\mathbb N$ such that $y=kx$.

In your case, $pq = q\cdot p$, which means that $p|q$ (because you can take $x=p, y=pq$ and $k=q$ in the definition.

For the same reason $p|pr$, so the third property is also true.

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The other two properties can be either true or false:

1. Property $1$ can be true (for example, if $p=2,q=2$) or false (if $p=2,q=3$ for example)
2. Property $2$ can be true (for example, $p=2, q=2, r=2$) or false ($p=2, q=2, r=3$).

Answer 2

The last statement is always true.

For the first three statements only, consider the following counter example: $$P=2, Q=3, R=9.$$