I'm studying number theory and I was given this Theorem to look at:
If $p \mid ab$ then $p \mid a$ or $p \mid b$
I had the following intuition for the problem or a proof of sorts if you will.
Intuitive Proof. We take the assumption that $p$ can in fact divide $ab$. Then, this implies that $p$ is divided out by one of prime factors of $ab$. The prime factors of $ab$ can be thought of as the union of the sets of primes factors of $a$ and $b$. This would mean that $p$ is in the set of prime factors of $a$ or $b$. More concretely:
Let $S$ be the set of primes factors of $ab$ which we defined to be the union of the sets of the prime factors of $a$ and $b$. For example:
Let $A$ be the set of primes factors of $a$ and let $B$ be the set of prime factors of $b$. Then $S=A \cup B$
Now, if $p \mid ab$ then $p$ must belong to $S$ ($p \in S$) this, therefore, implies that $p \in A$ or $p \in B$.
So, that's my "intuitive proof". Does it make sense?
All feedback is much appreciated.
Thanks a bunch!
EDIT: $p$ is prime.