Results 1 to 3 of 3

Math Help - [SOLVED] Dividing problem

  1. #1
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6

    [SOLVED] Dividing problem

    Hi !

    I've got a little problem, seems simple... Here it is :


    Text :
    We're going to study all couples (n,k) such as :

    {n \choose k}={n+1 \choose k-1}


    Question :
    Let p=n-k+1 and t=gcd(p,k). Also s=\frac{k}{t}

    Show that p=t^2


    So far... :
    By replacing the combinations with their formulas with !, i've found that :

    p(p+1)=(p+k)k

    p^2+p=pk+k^2

    So it's easy to show that t^2|p


    And now... ?
    I can't find how to show that p|t^2 or p \leq t^2 :'(

    I thought Bézout's identity could help, but no...


    Thanks in advance !
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Moo View Post
    I've found that :

    p(p+1)=(p+k)k
    p^2+p=pk+k^2

    So it's easy to show that t^2|p


    And now... ?
    I can't find how to show that p|t^2 or p \leq t^2 :'(
    So if t = \gcd(p,k), let p = rt and k = st, where r and s are co-prime.

    Then rt(rt+1) = t^2s(r+s), hence r(rt+1) = s(r+s)t.

    Since t has to divide the left-hand side of the last equation, and t is clearly co-prime with rt+1, it follows that t divides r (so that t^2 divides p: you already knew that).

    Similarly, since r has to divide the right-hand side of that equation, and r is co-prime with both s and r+s, it follows that r divides t.

    Put those two facts together, and you have r=t as required.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,

    Thanks for the answer ! This is exactly what i wanted (and even for the following question !)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Help dividing exponent by exponent
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 29th 2010, 11:20 AM
  2. Dividing
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 19th 2009, 11:07 PM
  3. [SOLVED] dividing by 0!
    Posted in the Statistics Forum
    Replies: 9
    Last Post: December 10th 2008, 07:50 PM
  4. [SOLVED] [SOLVED] Dividing Polymonials by Polynomials
    Posted in the Algebra Forum
    Replies: 1
    Last Post: March 7th 2008, 07:09 PM
  5. Problem of Dividing By Zero Solved
    Posted in the Math Forum
    Replies: 44
    Last Post: December 12th 2006, 04:05 AM

Search Tags


/mathhelpforum @mathhelpforum