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
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.