Results 1 to 2 of 2

Math Help - Euclidean division

  1. #1
    Junior Member
    Joined
    Aug 2009
    Posts
    62

    Euclidean division

    Hi everybody,

    I must show that the number a=n^2(n^2-1) is divisible by 12.

    I know just that (n-1).n.(n+1) is divisible by 6

    Do you have any idea?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,829
    Thanks
    1602
    Quote Originally Posted by lehder View Post
    Hi everybody,

    I must show that the number a=n^2(n^2-1) is divisible by 12.

    I know just that (n-1).n.(n+1) is divisible by 6

    Do you have any idea?
    If the number is divisible by 6, you know it must be divisible by 3.

    We can show that the number is also divisible by 4.

    If it is divisible by 4 and divisible by 3, it must be divisible by 12.


    Case 1: n is odd. Therefore n = 2m + 1.

    So n^2(n^2 - 1) = (2m + 1)^2[(2m + 1)^2 - 1]

     = (4m^2 + 4m + 1)(4m^2 + 4m + 1 - 1)

     = (4m^2 + 4m + 1)(4m^2 + 4m)

     = 4(4m^2 + 4m + 1)(m^2 + m).

    So the number is divisible by 4.


    Case 2: n is even. Therefore n = 2p.

    So n^2(n^2 - 1) = (2p)^2[(2p)^2 - 1]

     = 4p^2[4p^2 - 1].

    So the number is divisible by 4.


    Since the number is divisible by 3 and by 4, the number must be divisible by 12.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. non-euclidean
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: December 3rd 2010, 05:57 AM
  2. Euclidean help
    Posted in the Geometry Forum
    Replies: 1
    Last Post: July 22nd 2009, 10:42 AM
  3. Polynomial division vs synthetic division
    Posted in the Algebra Forum
    Replies: 2
    Last Post: April 9th 2009, 06:49 AM
  4. euclidean division
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: December 16th 2008, 11:46 AM
  5. Help w/ Euclidean Rings and P.I.D.s
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 31st 2008, 10:06 PM

Search Tags


/mathhelpforum @mathhelpforum