How can I prove that there is only one set of prime factors for a number?

Question

How can you prove that a number cannot have more than one set of prime factors?

You would know that $15$, for example, has $2$ prime factors $3$ and $5$.

you can easily know that these are the only prime factors because you can try dividing $15$ all the primes numbers less than $15$, and you wouldn't find any other number. But how can you know that for any number $n$, there cannot be more than one set of prime factors?

Another way to state this(after trying to prove my theory, I ended up with this)

if you have a number $A$ and a prime $p$ both not divisible by prime $q$, how can you prove that $A\cdot p$ is not divisble by $q$?

Answer 1

Well, the 2 numbers are prime, say $a$, $b$. Then our product is $ab$. Evidently, it has only 2 prime factors, since factors are only $a$ and $b$.

Answer 2

$A×p$ won't be divisible by $q$, because $p$ isn't... I saw this the other day, i believe it's called Euclid's lemma...

The fundamental theorem of arithmetic answers your question though...