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Math Help - relatively prime integers....

  1. #1
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    relatively prime integers....

    Hello everbody...

    Let b and c be relatively prime integers, and suppose a is an integer that is
    divisible by both b and c. Prove that bc|a.
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  2. #2
    Super Member Aryth's Avatar
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    If a is divisible by both b and c, we have:

    bk = a

    cj = a

    b = a/k

    c = a/j

    Therefore:

    bc = a/jk

    jk = i

    bc(i) = a

    Thus:

    bc | a
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  3. #3
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    Quote Originally Posted by Aryth View Post
    If a is divisible by both b and c, we have:

    bk = a

    cj = a

    b = a/k

    c = a/j

    Therefore:

    *** bc = a/jk***

    jk = i

    bc(i) = a

    Thus:

    bc | a
    ummm.... bc = a^2/jk. how does that work?
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  4. #4
    Senior Member JaneBennet's Avatar
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    b and c divide a \Rightarrow a=hb=kc for some integers h and k.

    b and c are relatively prime means there are integers p,q such that pb+qc=1.

    Now multiply through by a.

    pba+qca=a \Rightarrow a=pb(kc)+qc(hb)=(pk+qh)bc.
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