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Math Help - Dividable by 5

  1. #1
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    Dividable by 5

    I got to proof that
    n^5 - n
    is dividable by 5 for n >= 0, n in Natural numbers..
    And that by Induction..
    The "skelet" was succesfull, now I got to proof that
    P(i) => P(i+1)
    ..
    Any ideas?
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  2. #2
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    Quote Originally Posted by kirschplunder View Post
    I got to proof that
    is dividable by 5 for n >= 0, n in Natural numbers..
    And that by Induction..
    Say that K^5-K is divisible by five.
    Then expand this (K+1)^5-(K+1).
    After collecting terms, what do you see?
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  3. #3
    Senior Member Shanks's Avatar
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    This is the famous Fermat' small Theorem for the case n=5 in Number Theory.
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  4. #4
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    Quote Originally Posted by Plato View Post
    Say that K^5-K is divisible by five.
    Then expand this (K+1)^5-(K+1).
    After collecting terms, what do you see?
    I did that, then I made
    K^5 + 1^5 - K - 1 = K^5 - K
    But I didn't think that was really the right way?
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  5. #5
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    Quote Originally Posted by kirschplunder View Post
    I did that, then I made
    K^5 + 1^5 - K - 1 = K^5 - K
    But I didn't think that was really the right way?
    To do these you must understand basic algebra.
    (K+1)^5=K^5+5K^4+10K^3+10K^2+5K+1
    The basic operations are standing in your way.
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  6. #6
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    Quote Originally Posted by Plato View Post
    To do these you must understand basic algebra.
    (K+1)^5=K^5+5K^4+10K^3+10K^2+5K+1
    The basic operations are standing in your way.
    I'm a bit dumb xD Sowwy.

    So.

    K^5+5K^4+10K^3+10K^2+5K+1 - (K-1) - (K^5 - K)
    is
    5K^4+10K^3+10K^2+5KI got this by substracting P(K) from it, but I don't really get why.
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  7. #7
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    (K+1)^5-(K+1)=(K^5-K)+(5K^4+10K^3+10K^2+5K)
    The sum of two multiples of five is a multiple of five.
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  8. #8
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    Thank you alot

    Thank you, really
    I was thinking way too difficult, gotta get rid of that
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  9. #9
    Senior Member Shanks's Avatar
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    Suppose K leaves a remainder of r when divided by 5,
    Then K^5 has the same remainder as r^5.
    So what we need to do is Just check whether the conclusion hold when r take the value from {0,1,2,3,4}.
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