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Math Help - Number theory, divisibility, mathematical induction

  1. #1
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    Number theory, divisibility, mathematical induction

    Use mathematical induction to show that (n to the power 5) - n is divisible by 5 for every positive integer n.

    Sorry again for the expression. Thanks guys
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  2. #2
    Member Glaysher's Avatar
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    n = 1

    1^5 - 1 = 1 - 1 = 0

    Assume true for n = k

    k^5 - k = 5m for some integer m

    Suppose n = k + 1

    (k + 1)^5 - (k + 1) = k^5 + 5k^4 + 10k^3 + 10k^2 + 5k + 1 - k - 1

    = k^5 - k + 5(k^4 + 2k^3 + 2k^2 + k)

    = 5m + 5(k^4 + 2k^3 + 2k^2 + k)

    = 5(m + k^4 + 2k^3 + 2k^2 + k)

    Hence true for n = k + 1 and by the principle of mathematical induction true for all natural numbers n
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