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Math Help - Relitivly prime, unique divisors of divisors

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    Relitivly prime, unique divisors of divisors

    I was wondering if someone could help me prove the following.


    Let (a, b)=1. let d|ab. show that their exists unique d_1, d_2 such that d_1d_2=d,  d_1|a, d_2|b
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Chris11 View Post
    I was wondering if someone could help me prove the following.


    Let (a, b)=1. let d|ab. show that their exists unique d_1, d_2 such that d_1d_2=d,  d_1|a, d_2|b
    This makes no sense. (4,17)=1 and surely 4\mid 4\cdot 17 but d=4\cdot 1 and 4=2\cdots 2.
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    @Drexel28 - In your example, we have a = 4, b = 17 and d = 4. If we let d_1 = 4 and d_2 = 1, then it seems to me that we've met the requirements of the problem.

    @Chris11 - Are you allowed to use the Fundamental Theorem of Arithmetic (i.e., unique factorization) in your solution? If so, that's a hint.
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    Quote Originally Posted by Chris11 View Post
    Let (a, b)=1. let d|ab. show that there exist unique d_1, d_2 such that d_1d_2=d,  d_1|a, d_2|b
    You could start by letting d_1 = \gcd(a,d). Then d_1 divides  d, so let d_2 = d/d_1. Also, there exist integers x, y such that ax+dy = d_1. Multiply that equation by  b, and use the fact that ab is a multiple of  d to conclude that d_2 divides  b.

    For the uniqueness, you could try to show that d_1 must necessarily be equal to \gcd(a,d).
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