Thread: 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

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

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

Hello, Rimas!

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.