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Proof that LCM(c,d) divides LCM(a,b), known GCD(a,b) divides GCD(c,d)

  1. Nov 20, 2011 #1
    ** The problem statement, all variables and given/known data
    set S = set of the multiples of any two natural numbers a, b
    S = {n in N such that a|n and b|n}

    Denote min(S) = LCM(a,b) = least common multiple of a and b

    From previous result, I already proved that :
    If a divides c and if b divides d, then GCD(a,b) divides GCD(c,d)

    Now the question is to prove: LCM(a,b) divides LCM(c,d)

    ** My thoughts:
    By definition of set S, x = LCM(a,b) satisfies the fact that a|n and b|n for n in N
    I think, y = LCM(c,d) satisfies set S when c|m and d|m for m in N

    But after that, I get confused on what to do next.
    I think of trying to prove: LCM(c,d) = k * LCM(a,b), but I'm not sure if this is the right direction, or I need to do something else.

    Could someone please give me on hints on what to do? Thank you in advance.:smile:
  2. jcsd
  3. Nov 22, 2011 #2


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    Staff Emeritus
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    Hint: If a|n and b|n, then LCM(a,b)|n
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