Results 1 to 3 of 3

Math Help - sums and products...

  1. #1
    Newbie Lasombra's Avatar
    Joined
    Jul 2011
    From
    SouthEast suburbs of Athens
    Posts
    3

    sums and products...

    i suppose this is the correct place to post this question...it is easy to prove that 2 different pairs of Natural numbers cannot have the same sum AND the same product.Can this be proven for groups of 3,4,5,...n Natural numbers?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Sep 2011
    Posts
    58

    Re: sums and products...

    For 2 different pairs of natural numbers we can do it without much difficulty. Let our pairs be \{a,b\} and \{c,d\} and remember these sets are not equal.

    Suppose that ab=cd and a+b=c+d. Since a\not = c we can say a=c+k for some non-zero number k

    To ensure a+b=c+d we then know that b=d-k

    Then ab=(c+k)(d-k)=cd-ck+dk-k^2 and since ab=cd we know that we must have -ck+dk-k^2=0

    So by factoring we have k(d-c)-k^2 = k((d-c)-k)=0. Remember that k\not = 0 and so d-c-k=0 which means k=d-c

    Therefore a=c+k=c+d-c=d and b=d-k=d-(d-c)=c

    Therefore \{a,b\}=\{c,d\} but this is a contradiction. So two different pairs of natural numbers cannot have both the same sum and same product.
    Last edited by MonroeYoder; November 17th 2011 at 08:09 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7

    Re: sums and products...

    Quote Originally Posted by Lasombra View Post
    i suppose this is the correct place to post this question...it is easy to prove that 2 different pairs of Natural numbers cannot have the same sum AND the same product.Can this be proven for groups of 3,4,5,...n Natural numbers?
    It doesn't work for sets of four numbers. The sets {1,4,4,6} and {2,2,3,8} each have sum 15 and product 96.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Sums and products of random variables
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: May 24th 2011, 11:49 PM
  2. Probability: Sums and Products of Random Variables
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: May 12th 2011, 05:44 AM
  3. ...norm takes products to products.
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 23rd 2010, 09:55 PM
  4. Continuity of sums, products, and quotients of functions
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: January 28th 2010, 12:02 PM
  5. sums and products of roots
    Posted in the Algebra Forum
    Replies: 10
    Last Post: September 2nd 2009, 10:32 AM

Search Tags


/mathhelpforum @mathhelpforum