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Math Help - Don't know how to proof it

  1. #1
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    Don't know how to proof it

    n and k are natural numbers, k is odd

    Prove that ( 1 + 2 + 3 + ... + n ) is a factor of ( 1k + 2k + 3k + ... + nk )

    I start the proof by using mathematical induction ( but don't know how to do it ), then I try to tackle from a special case :
    [ e.g. The case ( 1 + 2 ) is a factor of ( 1k + 2k ) where k is natural and odd --- I can proof it but do not know how to extend it ]

    Can anyone give me a suggestion ( Better be simple because I am not good at math. )



    ** This is not a homework. I saw this question on another web site ( not a math. forum ) 2 days ago.
    I thought it was an easy question so I have spent the past 2 days to solve it. ( It seems I was overly confident )
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  2. #2
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    Re: Don't know how to proof it

    Oh.. I thought this was your homework. The good thing that you have a passion to help whoever that person, but I think I'm late to answer this one. Sorry dude.
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  3. #3
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    Re: Don't know how to proof it

    Quote Originally Posted by JeanGunter View Post
    Oh.. I thought this was your homework. The good thing that you have a passion to help whoever that person, but I think I'm late to answer this one. Sorry dude.
    JeanGunter, you have repeatedly posted a response saying you cannot help! Why? There are, I am sure, millions of people who cannot help with a particular problem. If they all posted to say so, this board would be overwhelmed!

    AnEducatedMonkey, you can't prove it. It is NOT true. In particular, if n= 2 and i= 2, this asserts that 1+ 2= 3 must divide 1^2+ 2^2= 1+ 4= 5 and that obviously is not true.
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    Re: Don't know how to proof it

    Quote Originally Posted by HallsofIvy View Post
    JeanGunter, you have repeatedly posted a response saying you cannot help! Why? There are, I am sure, millions of people who cannot help with a particular problem. If they all posted to say so, this board would be overwhelmed!

    AnEducatedMonkey, you can't prove it. It is NOT true. In particular, if n= 2 and i= 2, this asserts that 1+ 2= 3 must divide 1^2+ 2^2= 1+ 4= 5 and that obviously is not true.
    he does specify k (what I think you mean by i) is odd. It does seem to work for k odd. I took a quick look. Unless I missed something it's a non-trivial problem. Specifying the sum of 1 to n is simple enough, but for general odd k the second sum involves the generalized Harmonic number (over my head). Info can be found by googling "power sum".
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    Re: Don't know how to proof it

    1+2 divides 13+23
    (1+2)(22+a)=22+a+23+2a=1+23
    a=-1
    ck:
    (1+2)(22-1)=13+23

    1+2+3 divides 13+23+33
    (1+2+3)(22+32+a)= 1+23+33
    solve for a. That’s the pattern:

    (1+2+3) divides 15+25+35
    (1+2+3)(24+34+a)= 1+25+35
    solve for a.

    etc, etc
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