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Math Help - Least common multiple - Greatest common divisor

  1. #1
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    Least common multiple - Greatest common divisor

    So I have 2 positive integers a and b, a<b.
    And their LCM minus GCD is 143.
    Find a and b.
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  2. #2
    Member HappyJoe's Avatar
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    The problem is just to find any pair (a,b) for which LCM(a,b) - GCD(a,b) = 143?

    You can take a=1, b=144. Their GCD is 1, while their LCM is 144.
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  3. #3
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    No, the problem is to find all pairs, forgot to mention that sry.
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  4. #4
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    do you know the answer - because i think there are infinate answers - or have i been away from amths for so long im making stupid errors?
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    LCM(a,b) - GCD(a,b) = 143

    LCM(a,b)=k*GCD(a,b), k is some integer.

    LCM(a,b) - GCD(a,b) = GCD(a,b) {k-GCD(a,b)}=143

    hmmm...
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  6. #6
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    Quote Originally Posted by Also sprach Zarathustra View Post
    LCM(a,b) - GCD(a,b) = 143

    LCM(a,b)=k*GCD(a,b), k is some integer.

    LCM(a,b) - GCD(a,b) = GCD(a,b) {k-GCD(a,b)}=143

    hmmm...
    You mean LCM(a,b) - GCD(a,b) = GCD(a,b) (k-1)=143=13*11, thank you from here on i make 4 cases, and get that:
    (a,b)={(22,77);(11,154);(39,52);(13,156);(9,16);(1 ,144);(143,246)}
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  7. #7
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by teps View Post
    You mean LCM(a,b) - GCD(a,b) = GCD(a,b) (k-1)=143=13*11, thank you from here on i make 4 cases, and get that:
    (a,b)={(22,77);(11,154);(39,52);(13,156);(9,16);(1 ,144);(143,246)}
    Typo, thanks...

    Good-job btw.
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