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Math Help - gcd(a,b) proof - need help

  1. #1
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    gcd(a,b) proof - need help

    Hello and thanks for looking

    I have to prove the following :Prove that gcd(a,b) = 1 if and only if gcd(ab,a+b)=1


    Thanks a lot in advance for the help
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  2. #2
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    Quote Originally Posted by octagonreturns View Post
    Hello and thanks for looking

    I have to prove the following :Prove that gcd(a,b) = 1 if and only if gcd(ab,a+b)=1


    Thanks a lot in advance for the help
    Let d=\gcd(ab,a+b) then d|ab so d|a or d|b. WLOG say d|a then since d|(a+b) it means d|(a+b-a)\implies d|b. And so d=1.
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  3. #3
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    Hello,

    Proving this implication : gcd(ab,a+b)=1 \implies gcd(a,b)=1

    Let d=gcd(a,b)

    d divides a and d divides b. Therefore, d (and even d˛) divides ab and d divides (a+b).

    But we know that gcd(ab,a+b)=1.

    --> d divides 1.

    So d=1.
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    Thank you all so very much for your time and help!
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