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Math Help - Induction Proof

  1. #1
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    Induction Proof

    Prove that for all n \geq 1, prove 5 divides 8^n - 3^n using inducation.

    ----

    So I know the inductive step:

    Assume 5 divides 8^k-3^k

    Show 5 divides 8^{k+1} - 3^{k+1}

    So, 8^k - 3^k = 5p, for some integer p. This is the inductive hypothesis.

    8^{k+1}-3^{k+1} = \left(8^k\cdot 8 - 3^k\cdot 3\right) = \left(8^k\cdot (5+3) - 3^k\cdot 3\right) = \left(5\cdot 8^k + 3\cdot 8^k\right) - 3^k\cdot 3

    I'm stuck from here. Thanks for the help.
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  2. #2
    Member arpitagarwal82's Avatar
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    8^{k+1}-3^{k+1} = \left(8^k\cdot 8 - 3^k\cdot 3\right)

    8(8^k - 3^k) + 5.3^k
    = 5p + 5.3^k
    = 5( p + 3^k)
    hence divisible by 5.
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