What is the largest possible product of a collection of positive real numbers which sums to exactly 27?
the product is largest when the numbers are about equal. Just as a square has the largest area for a given perimeter.
13*14=182
9*9*9=721
6*7*7*7=2058
6*5*5*4*4=2400
You can see that the more numbers you have, the greater the product. So, n times the nth root will keep increasing with larger n:
(27/2)^2 = 182.25
(27/3)^3 = 721.00
(27/4)^4 = 2075.94
(27/5)^5 = 4591.65
(27/6)^6 = 8303.76
So, I guess if you want integers, the maximum product will be
1*2^13 = 8192
3*2^12 = 12288 <---
5*2^11 = 10240