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March 28, 2017

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Find the largest integer N for which N−6 evenly divides N^3−6.

  • Math (Algebra) - ,

    N^3−6 is zero Modulo (N-6). If we compute Modulo (N-6) then obviously:

    N-6 = 0 ----->

    N = 6

    Here and in the following the equals sign means equality modulo N - 6.

    We then have:

    N^3 -6 = 6^3 - 6 = 210

    Therefore:

    210 = 0

    Reverting back to the ordinary definition of the equals sign, this means that:

    210 = k (N-6)

    So, N-6 must be the largest possible factor of 210, which is 210 therefore
    N = 216.

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