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Math Help - Relationships between Highest Common Factor and Lowest Common Multiple?

  1. #1
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    Exclamation Relationships between Highest Common Factor and Lowest Common Multiple?

    Why is the product of an original pair of numbers same as the product of their Highest Common Factor and Lowest Common Multiple?

    For example,
    HCF of 12 and 16 = 4
    LCM of 12 and 16 = 48

    12 x 16 = 192
    4 x 48 = 192

    Please explain as clearly as possible. Thank You.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Taguig City, Philippines
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    Quote Originally Posted by pepperhu View Post
    Why is the product of an original pair of numbers same as the product of their Highest Common Factor and Lowest Common Multiple?

    For example,
    HCF of 12 and 16 = 4
    LCM of 12 and 16 = 48

    12 x 16 = 192
    4 x 48 = 192

    Please explain as clearly as possible. Thank You.
    because it is a theorem..
    if x, y are positive numbers, then lcm(x,y) hcf(x,y) = xy..

    you may want to read the explanation (proof) here..
    Basic Number Theory: LCM/GCD Proof
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  3. #3
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    Hello, pepperhu!

    Why is the product of an original pair of numbers
    same as the product of their HCF and LCM?

    For example,
    HCF of 12 and 16 = 4
    LCM of 12 and 16 = 48

    12 x 16 = 192
    4 x 48 = 192

    Please explain as clearly as possible.

    Here's an informal explanation.


    The HCF contains the "overlap" of their factors.

    The LCM contains their factors minus their "overlap".

    Together (multiplied), they contain all their factors.

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