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Math Help - prime problem

  1. #1
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    Question 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
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  2. #2
    Senior Member Dinkydoe's Avatar
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    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.
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  3. #3
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    That's what I thought
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