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:

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- September 30th 2006, 09:16 AMRimasMaximized 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: - October 1st 2006, 01:33 AMexcelmaths.comMaximum 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 - October 1st 2006, 02:54 AMCaptainBlack
- October 1st 2006, 05:42 AMSoroban
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**.