Results 1 to 3 of 3

Math Help - [SOLVED] difference of two squares problem!!

  1. #1
    Newbie nandu11's Avatar
    Joined
    Apr 2008
    From
    earth..
    Posts
    3

    [SOLVED] difference of two squares problem!!

    the question is that K(N) denotes the no. of ways in which N can be expressed as the difference of two perfect squares, then which of the following is maximum: K(110), K(105), K(216), K(384)????
    i need to know where to start from.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2007
    Posts
    329
    I think you should start from: a^2-b^2=(a-b)(a+b). So you need to factor those numbers...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,659
    Thanks
    600
    Hello, nandu11!

    Lucky for us, I've played with the Difference-of-Squares long ago.


    K(n) denotes the number of ways in which n
    can be expressed as the difference of two perfect squares.

    Then which of the following is maximum? . K(110),\;K(105),\;K(216),\;K(384)

    Suppose n is a product, PQ, where P \geq Q.

    Then we want integers a and b so that: . a^2-b^2 \:=\:PQ

    So we have: . (a+b)(a-b) \:=\:PQ

    We assume that: . \begin{array}{ccc}a+b &=&P \\ a-b &=& Q\end{array}

    . . and solve the system: . \begin{array}{ccc}a &=&\dfrac{P+Q}{2} \\ \\[-4mm] b &=&\dfrac{P-Q}{2}\end{array}


    Since a and b are integers, P and Q must have the same parity.
    . . (Both are even or both are odd.)



    Let's examine each of the n's.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    n \:=\: 110 \:=\:2\cdot5\cdot11

    110 cannot be factored into two factors with the same parity.
    Hence, 110 cannot be expressed as a difference of squares.
    . . K(110) \:=\:0


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    n \:=\:105 \:=\:3\cdot5\cdot7

    105 has four possible factorings.

    . . \begin{array}{c|c}<br />
(P,Q) & (a,b) \\ \hline<br />
(105,1) & (53,52) \\ (35,3) & (19,16) \\ (21,5) & (13,8) \\ (7,15) & (11,4) \end{array}\quad K(105)\:=\:4


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    n = 216 \:=\:2^3\cdot3^3

    216 has four possible factorings.

    . . \begin{array}{c|c}<br />
(P,Q) & (a,b) \\ \hline<br />
(108,2) & (55,53) \\ (54,4) & (29,25) \\ (36,6) & (21,15) \\ (18,12) & (15,3) \end{array} \quad K(216) \:=\:4


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    n \:=\:384 \:=\:2^7\cdot3

    384 has six possible factorings.

    . . \begin{array}{c|c}<br />
(P,Q) & (a,b) \\ \hline<br />
(192,2) & (97,95) \\ (96,4) & (50,46) \\ (64,6) & (35,29) \\ (48,8) & (28,20) \\ (32,12) & (22,10) \\ (24,16) & (20,4) \end{array} . K(384) \:=\:6 . {\color{blue}\leftarrow\text{ maximum!}}

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Difference of two squares
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 22nd 2010, 09:05 AM
  2. difference of squares
    Posted in the Algebra Forum
    Replies: 1
    Last Post: November 18th 2009, 02:37 AM
  3. [SOLVED] Difference of two Squares
    Posted in the Algebra Forum
    Replies: 3
    Last Post: March 24th 2009, 08:02 PM
  4. Difference of Squares
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 4th 2008, 06:40 PM
  5. [SOLVED] difference of two squares
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: July 7th 2007, 11:02 AM

Search Tags


/mathhelpforum @mathhelpforum