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

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 $q$?

• Commented Jan 26, 2018 at 8:38
• What you are speaking of is called [the fundamental theorem of arithmetic][1]. As the name hints, the theorem is not trivial to prove. [1]: en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic
– 5xum
Commented Jan 26, 2018 at 8:39

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

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 $b$.

• But how do you know the factors are unique? (That's what OP is asking, after all)
– 5xum
Commented Jan 26, 2018 at 8:40
• The number ab where a and b are primes has evidently 2 factors only(excluding 1 and itself) Commented Jan 26, 2018 at 8:45
• Without already knowing the fundamental theorem of arithmetic (which OP clearly doesn't know), how do you know that?
– 5xum
Commented Jan 26, 2018 at 8:53
• Factoring. That way is rather simple Commented Jan 26, 2018 at 9:06
• But how do you know factoring is unique? How do you know that if $p_1p_2=q_1q_2$ where $p_i,q_i$ are prime, then $\{p_1,p_2\}=\{q_1,q_2\}$?
– 5xum
Commented Jan 26, 2018 at 9:07