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Math Help - Simple Proofs Problem

  1. #1
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    Simple Proofs Problem

    I need to show that (n^2) - 1 is a multiple of 24. n is an odd number but not divisible by 3.

    I have a general understanding of the problem, but i am getting tripped up on the not divisible by 3 part.

    Any help would be much appreciated!
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  2. #2
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    Hello, dlee426!

    Prove that N \:=\:n^2 - 1 is a multiple of 24,
    where n is an odd number but not divisible by 3.

    We have: . n is of the form 3a \pm 1, where a is even.
    . . That is, a = 2b for some integer b.

    Then: . n \:=\:3(2b) \pm 1 \:=\:6b\pm 1

    And: . N \:=\:n^2-1 \:=\:(6b\pm1)^2 - 1 \:=\:36b^2 \pm12b \:=\:12b(b\pm1)


    We see that N is divisible 12.

    Note that b(b\pm1) is the product of two consecutive integers.
    . . That is, one of them is even, the other odd.
    Hence, the product b(b\pm1) is divisible by 2.


    Therefore, N is divisible by 24.

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