# Math Help - prime problem

1. ## prime problem

This is a question that interests me and I don't know how to find the answer: Do numbers exist that are simultaneously the product of two different pairs of primes? Can you give examples or proof?

I mean, can Pa * Pb = Pc * Pd?

TIA

2. No, primefactorisation is unique. A statement like $p_ap_b = p_cp_d$ is a contradiction. If $n=m$, then $n,m$ can be divided by the same primes. And so eventually $n,m$ get the same primefactorisation. You can't find a prime that divides $n$, that does not divide $m$. It's really trivial

If $p_1\cdots p_n= n$. And suppose there's another element $p$, which is not in the factorisation of $n$, but does divide $n$. Then $p|p_i$ for some prime. But then $p=p_i$ or $p=1$.

3. That's what I thought