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Math Help - Number Theory GCD

  1. #1
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    Number Theory GCD

    let a,b,c in Z with (a,b)=1. If a divides c and b divides c, then ab divides c.
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  2. #2
    Moo
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    Hello,

    This one is "simple" ^^

    If a divides c, then one can write c=ka where k is an integer.
    Similarly, one can write c=k'b where k' is also an integer.

    Thus we have the equality ka=k'b.
    So we can deduce that b divides ka.
    Since b and a are coprime, b divides k. *this can be proved, see below
    So one can write k=k''b where k'' is an integer.

    Substitute this in c :
    c=ka=k''ba

    Hence ab divides c

    ----------------------------
    * proof :
    let p be a divisor of b.
    Since it divides b, it divides ka. But it cannot divide a otherwise gcd(a,b) wouldn't be 1. Therefore it divides k.
    So for any divisor of b, it divides k. That is to say b divides k.
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