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Math Help - Prime Factorization

  1. #1
    Junior Member
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    Prime Factorization

    Here is what I need to prove:

    Let a & b be positive integers, and let m be an integer such that ab=m(a,b). Without using the prime factorization theorem, prove that (a,b)[a,b] = ab by verifying that m satisfies the necessary properites of [a,b].

    I know that the necessary properties of [a,b] are
    let a,b be positive integers. The LCM is a positive integer m such that,
    (1) a|m & b|m
    (2) for any n with a|n & b|n, we must have m|n

    Any suggestions on proving this.
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  2. #2
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    If you can prove that (a,b)=1 \Rightarrow [a,b]=ab and [da,db] = d[a,b] then

    it will be short to get (a,b)[a,b] = ab
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  3. #3
    MHF Contributor
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    Let p be the prime factors of a and b.

    gcd(a,b)=p_1^{min \ a,b}*....p_k^{min \ a,b}
    gcd(a,b)=p_1^{max \ a,b}*....p_k^{max \ a,b}

    What happens when you multiple the prime factorization together?
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