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p|n => pa=n for some integer a
q|n => qb=n for some integer b
we want to show pqk=n for some integer k
we also have that p and q are two distinct primes
if we were to give the prime factorization for n it would look something like p*q*some other prime integers possibly
And we know that a and b have prime factorizations as well, so we could say that a and b are some product of primes and then that n^2 is pq times some other primes, but how do we know that pq isn't larger than n?
Oh, is it because if p and q both divide n then they are both part of n? Kinda an Euler's Totient type thing?
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Yeah since both p and q divide n then we know the prime factorization for n=p*q(some other primes possibly)
2|972 => 2a=972
3|972 => 3b=972
But as we can see in the prime factorization we also have 2*3*k=n
Thanks, that makes a lot of sense. Much more clear now.
IF you want a more rigorous way you can do the following:
since p and q are distinct primes then theie gcd is 1 and so they can be written as a linear combination
\[\alpha p+\beta q=1\]
and using the notation from above \(n=ap\) and \(n=bq\)
\[n=n\cdot 1=n(\alpha p+\beta q)\]
\[=n\alpha p+n\beta q=bq\alpha p+ap\beta q\]
so \(n\) is an integer multiple of \(pq\). Thus