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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 $q$?

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2 Answers 2

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$A×p$ 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...

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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 $b$.

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

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