# Maximized Product with given sum

• Sep 30th 2006, 08:16 AM
Rimas
Maximized Product with given sum
Amoung all collevtions,S, of positive itegers whose sum is 28 what is the largest product that the intergers in s can form

My teacher gives problems that she never teaches in school:confused:
• Oct 1st 2006, 12:33 AM
excelmaths.com
Maximum value of product
Collection S={ x,y are integers such that x,y>0 and x+y=28}

Let P be the product of x and y => P=xy
Now we have y=28-x => P=x.(28-x)
=28x-(x^2)

To find the largest value of the product P, differentiate P with respect to x and equate to zero.
We get 28-2x=0
=> 28=2x
=> x=14
thus y=28-x=14
So the largest product =x.y=196
• Oct 1st 2006, 01:54 AM
CaptainBlack
Quote:

Originally Posted by excelmaths.com
Collection S={ x,y are integers such that x,y>0 and x+y=28}

Let P be the product of x and y => P=xy
Now we have y=28-x => P=x.(28-x)
=28x-(x^2)

To find the largest value of the product P, differentiate P with respect to x and equate to zero.
We get 28-2x=0
=> 28=2x
=> x=14
thus y=28-x=14
So the largest product =x.y=196

The question does not restrict the size of S to two elements (it may have
intended to but it does not).

RonL
• Oct 1st 2006, 04:42 AM
Soroban
Hello, Rimas!

Quote:

Among all collections, S, of positive integers whose sum is 28
what is the largest product that the integers in S can form?

The Captain is correct: the number of integers is not specified
and the problem appears to be deliberately worded that way.

It can be shown that, given a set of numbers with a fixed sum,
. . the maximum product is achieved when the numbers are equal.

Let n = number of integers. .Then each number is: 28/n
. . Their product is: .P .= .(28/n)^n

Since 28/n is an integer, n must be a divisor of 28.
There are only six of them . . .

n = 1: . P .= .28^1 .= . . .1

n = 2: . P .= .14^2 .= . .196

n = 4: . P .= . 7^4 . = . 2,401

n = 7: . P .= . 4^7 . = .16,384 .

n = 14: .P .= .2^14 .= .16,384 .

n = 28: .P .= .1^28 .= . . .1

Therefore, the largest possible product is 16,384.