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Math Help - Prove problem

  1. #1
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    Question Prove problem

    Hi,
    Could someone please explain how to do this:

    Prove that n^5 - n is divisible by 10 for all positive integer n.
    [Hint: a number is divisible by 10 if and only if it is divisible by 5 and ....]

    thanks!
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  2. #2
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    Quote Originally Posted by quah13579 View Post
    Hi,
    Could someone please explain how to do this:

    Prove that n^5 - n is divisible by 10 for all positive integer n.
    [Hint: a number is divisible by 10 if and only if it is divisible by 5 and ....]

    thanks!
    (your hint is incorrect with the double implication, while it is true that if a number is divisible by 10 then it is divisble by 5, it is not true to say that if a number is divisible by 5 then it is divisible by 10 for ex:25, though I'm guessing a 2 shoulda been after that "and")

    Proof by induction

    Base case n=1, 1^5-1=1-1=0 and 0 is divisible by 10 since 0=10*0
    (an integer a is divisible by an integer b if there exists some c such that a=bc)


    Now assume n^5-n is divisible by 10... which means n^5-n=10k for some k\in\mathbb{Z}

    Consider (n+1)^5-(n+1)=n^5+5n^4+10n^3+10n^2+5n+1-n-1=n^5-n+5n^4+10n^3+10n^2+5n (we just rearranged here and cancelled the 1's)

    By our assumption we can substitute in for the first part of the expression so

    =10k+5n^4+10n^3+10n^2+5n=10(k+n^3+n^2)+5(n^4+n)

    Now show n^4+n is divisible by 2 for all positive integers, then the 5*2 will give us a 10 and the entire expression will be divisible by 10

    Can you do this induction?
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  3. #3
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by quah13579 View Post
    Hi,
    Could someone please explain how to do this:

    Prove that n^5 - n is divisible by 10 for all positive integer n.
    [Hint: a number is divisible by 10 if and only if it is divisible by 5 and ....]

    thanks!
    n^5-n is divisible by 5 by Fermatís little theorem. It is also even (as n^5 and n are either both even or both odd). Hence n^5-n is divisible by 10.
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