hey need help with sum maths hw?

Question

the problem is
the sum of n positive integers is 19. what is the maximum possible product of these n numbers?

anyone hav anbyideads how to do it?

Answer 1

I suppose the answer is 3 x 3 x 3 x 3 x 3 x 2 x 2 = 972.

In problems like these, the best you can do is having a maximum number of 3's, which has to do with 3 being close to Euler's number e. I don't have any theoretical basis for this, though.

Answer 2

You want to go exponential for this, so try to get the highest number as many times as you can. You can get 8 twos and a three, 5 threes and a four, 4 fours and a three, 3 fives and a four, etc. Looking at those, the threes and a four seem the best way to go, as you have the most numbers that are close together. This case gets you 972.

Answer 3

3+4+5+7 = 19
3*4*5*7 = 420

Answer 4

off the top of my head id say 15*14=210

only reason i come up with this is cuz products get bigger the larger the numbers being multiplied not the number of numbers, so thats would be my guess

Answer 5

the largest possible product is when the x's are all equal, so each x = 19/n
n product = (19/n)^n is the max

Answer 6

let x = one number
19 - x = another number
prod = x(19 - x) . . . . differentiate
prod ' = 19 - 2x = 0
x = 19/2 = 9.5
y = 9.5
max product = 9.5(9.5) = 90.25