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Math Help - Least Common Multiple

  1. #1
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    Least Common Multiple

    Could someone help me prove the following:

    I'm denoting the least common multiple as [a,b].

    (a) whenever a divides k and b divides k, then [a,b] divides k.
    (b) [a,b]=(ab)/(gcd(a,b)) if a,b are greater than zero.

    Thanks for the help.

    Jimmy
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  2. #2
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    Quote Originally Posted by JimmyT View Post
    Could someone help me prove the following:

    I'm denoting the least common multiple as [a,b].

    (a) whenever a divides k and b divides k, then [a,b] divides k.
    You know that there exists two integers a and x such that: ax=k

    You know that there exists two intergers b and y such that: by=k

    Let us say that the least common multiple of a and b is n

    Thus there exists two integers n and c such that: nc=a

    Thus there exists two integers n and d such that: nd=b

    Go back to: ax=k

    thus: x=k/a

    then: x=k/(nc)

    multiply both sides by c: xc=k/n

    Now, two integers multiplied together equal an integer, so k/n is an integer, thus k is divisible by n.
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  3. #3
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    Thank you so much.

    Can anyone help me with (b)?
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