x,y,& z are integers. Write a pf by contraposition to show that if 8 does not divide x^2-1, then x is even?

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  • 1 decade ago
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    If x is odd, then either

    x = 4k + 1 or

    x = 4k + 3

    for some integer k.

    If x = 4k + 1, then

    x^2 - 1 = (4k + 1)^2 - 1

    = 16k^2 + 8k = 8(2k^2 + k)

    is clearly divisible by 8.

    If x = 4k + 3, then

    x^2 - 1 = (4k + 3)^2 - 1

    = 16k^2 + 24k + 8 = 8(2k^2 + 3k + 1)

    is clearly divisible by 8.

    Therefore, if x is odd, then 8 divides x^2 + 1.

    Therefore, if 8 does not divide x^2 - 1, then x is even. (That's the contrapositive of the proven previous statement.)

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