Results 1 to 4 of 4

Math Help - Bimonial theorem and expansion

  1. #1
    Senior Member
    Joined
    Jan 2009
    Posts
    381

    Bimonial theorem and expansion

    Two positive integers , p and q are connected by p=q+1 . By using the binomial expansion , show that the expression p^{2n}-2nq-1 can be divided exactly by q^2 for all positive integers n .

    I can do this part .

    This is the continuation :

    hence(2) show that 3^{15}+5 can be divided exactly by 4 .


    i am not sure about this part .
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1

    Divisibility

    Hello thereddevils
    Quote Originally Posted by thereddevils View Post
    Two positive integers , p and q are connected by p=q+1 . By using the binomial expansion , show that the expression p^{2n}-2nq-1 can be divided exactly by q^2 for all positive integers n .

    I can do this part .

    This is the continuation :

    hence(2) show that 3^{15}+5 can be divided exactly by 4 .


    i am not sure about this part .
    It certainly looks as if we've got to put p = 3 and q = 2 here. This gives 3^{2n}-4n - 1 is divisible by 4.

    Obviously, we can't put n = 7.5, to give 3^{15} directly. But what about n = 8? This gives 3^{16} - 33 is divisible by 4. And 33 has 3 as a factor ...

    Can you see what to do next?

    Grandad
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Jan 2009
    Posts
    381
    Quote Originally Posted by Grandad View Post
    Hello thereddevilsIt certainly looks as if we've got to put p = 3 and q = 2 here. This gives 3^{2n}-4n - 1 is divisible by 4.

    Obviously, we can't put n = 7.5, to give 3^{15} directly. But what about n = 8? This gives 3^{16} - 33 is divisible by 4. And 33 has 3 as a factor ...

    Can you see what to do next?

    Grandad

    Thanks Grandad . I know that 3^{16}-33 is divisible by 4 but i still can't get it . Really sorry bout that . I guess i need more explaination .
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1

    Divisibility

    Hello thereddevils
    Quote Originally Posted by thereddevils View Post
    Thanks Grandad . I know that 3^{16}-33 is divisible by 4 but i still can't get it . Really sorry bout that . I guess i need more explaination .
    3^{16} - 33 = 3(3^{15} - 11), which is divisible by 4.

    So, since 3 isn't divisible by 4, 3^{15}- 11 is divisible by 4

    \Rightarrow 3^{15} - 11 + 16 is divisible by 4

    Grandad
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Quadratic Expansion using Binomial Theorem
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 8th 2012, 05:02 PM
  2. Replies: 12
    Last Post: July 26th 2010, 03:33 AM
  3. Binomial theorem expansion.
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: July 7th 2009, 11:15 PM
  4. Replies: 6
    Last Post: May 1st 2009, 11:37 AM
  5. [SOLVED] Trinomial expansion theorem
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: September 2nd 2007, 07:59 AM

Search Tags


/mathhelpforum @mathhelpforum