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Math Help - Gcd

  1. #1
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    Gcd

    How do you prove if positive integers a,b,c that if gcd(a,b)=1 and a/bc then a/c (without use of Fund. Theorem of Arithmetic)
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  2. #2
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    Quote Originally Posted by chadlyter View Post
    How do you prove if positive integers a,b,c that if gcd(a,b)=1 and a/bc then a/c (without use of Fund. Theorem of Arithmetic)
    If gcd(a,b)=1 there exists x,y in Z such that,

    ax+by=1 (1)

    a|bc thus there exists k in Z such that bc=ak. (2)

    Multiply (1) by c to get,
    acx+bcy=c
    Substitute (2) to get,
    acx+aky=c
    a(cx+ky)=c
    Thus, a|c.
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