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Thread: Number Theory (8)

  1. #1
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    Number Theory (8)

    Let m and n be positive integers with (m,n) =1. Prove that each divisor d>0 of mn can be written uniquely as d1d2 where d1,d2>0, d1 divides m, d2 divides n, and (d1,d2) =1 and each such product d1d2 corresponds to a divisor d of nm.
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  2. #2
    Super Member PaulRS's Avatar
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    Let $\displaystyle d|(m\cdot{n})$ and $\displaystyle d=d_1\cdot{d_2}=d_3\cdot{d_4}$ where $\displaystyle d_1|m,d_3|m$ and $\displaystyle d_2|n,d_4|n$

    we have: $\displaystyle
    d_1 \cdot d_2 = \dot d_3
    $ and since $\displaystyle (d_2,d_3)=1$ ( because $\displaystyle (n,m)=1$ ) it follows that $\displaystyle d_1=\dot d_3 $ (1)

    Again $\displaystyle
    d_3 \cdot d_4 = \dot d_1
    $ and since $\displaystyle (d_4,d_1)=1$ it follows that $\displaystyle d_3=\dot d_1 $ (2)

    By (1) and (2) we must have (think about the quotient) $\displaystyle d_1=d_3$ and from there $\displaystyle d_2=d_4$ which proves that the representation is unique
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