Results 1 to 3 of 3

Math Help - Greatest Product

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    1

    Question Greatest Product

    I'm working on this problem. I have an arbitrary number n. Then I'm finding out which will give me the biggest product when the numbers that make up the sum of n are multiplied together. The answer always seems to be a combination of 2's and 3's, but why is this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    May 2008
    Posts
    33
    suppose for the integer n you have \sum_{i=1}^k a_i = n , a_i>0. if one of them is larger than 3, lets say a_1 \geqslant 4 then a_1 = 2 + (a_1-2) where a_1 - 2 \geqslant 2 so  2*(a_1 - 2) = (a_1 - 2) +(a_1 - 2) \geqslant a_1 - 2 + 2  = a_1 and finally you get
    2 + (a_1 - 2) +  \sum_{i=2}^k a_i = n and  2*(a_1-2)*\prod_{i=2}^k a_i \geqslant \prod_{i=1}^k a_i
    (notice that the inequality is strict if the integer is greater than 4)
    so unless you only have 2's and 3's you can always get a better product

    you can show in a similar way that you should prefer 3's over 2's
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    You can read this.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding the greatest product
    Posted in the Algebra Forum
    Replies: 4
    Last Post: November 9th 2011, 06:06 AM
  2. greatest integers
    Posted in the Algebra Forum
    Replies: 2
    Last Post: August 23rd 2010, 08:43 AM
  3. The greatest value
    Posted in the Algebra Forum
    Replies: 1
    Last Post: February 22nd 2010, 06:39 PM
  4. greatest value
    Posted in the Algebra Forum
    Replies: 4
    Last Post: September 3rd 2009, 07:31 AM
  5. Greatest product from a sum
    Posted in the Number Theory Forum
    Replies: 8
    Last Post: July 28th 2007, 06:14 AM

Search Tags


/mathhelpforum @mathhelpforum