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Thread: Divisibility

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    Divisibility

    There are given certain positive integer numbers m,n,d. Prove that if numbers m^2n + 1 and mn^2 + 1 are divisible by d, then numbers m^3 + 1 and n^3 + 1 are also divisible by d.
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    I don't have an answer, but for forum organization purposes this question was asked a week ago by someone else.

    http://www.mathhelpforum.com/math-he...ty-154933.html
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  3. #3
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    Quote Originally Posted by PaulinaAnna View Post
    There are given certain positive integer numbers m,n,d. Prove that if numbers m^2n + 1 and mn^2 + 1 are divisible by d, then numbers m^3 + 1 and n^3 + 1 are also divisible by d.
    Just play around with multiples of d and the result will fall out:

    m^2n+1 = pd,\quad mn^2+1 = qd,

    m^2n^2 +n = npd,\quad m^2n^2+m = mqd,

    n-m = (np-mq)d, and so n = m+(np-mq)d,

    m^2\bigl(m+(np-mq)d\bigr) + 1 = pd,

    m^3+1 = (p-m^2np+m^3q)d.
    Last edited by Opalg; Sep 8th 2010 at 11:05 AM. Reason: corrected error
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