The sum of n positive integers is 19. What is the maximum possible product of these n numbers?
Can you also please explain why this is so?
The arithmetic-geomentric mean inequality tells us that if there are numbers involved, and that these are , then:
Regarding the lefthand side of this inequality as a function of N this has a maximum when . So we have that the maximum posible value of the product of an integer partition of is less than or equal .
The partition has product and I would not be supprised if this were the maximum, but have not done enough to convince even myself that this is the case.
The above example of a 7-partition with a product of shows that the partition that gives the maximum product must be a 6, 7 or 8-partition.
(Exhaustive search shows that is indeed the maximum product and it can be achived with either a 6 or a 7-partition.
The 6-partition of 19 which achives the maximum product can also be found im this way.
What about [19/(19/e)] to the power of (19/e)?
This gives 1085.405992...
I know the OP wanted integers, but this suggests the following procedure.
If the number leaves a remainder of 2 when divided by three, then the max. answer is 3*3*3*...*2.
If the number leaves a remainder of 1 when divided by three, then the max. answer is 3*3*3*...*2*2.
If there is no remainder on division by three, then it's 3*3*3*...*3.
E.g. for 23 it's 3^7 * 2, and for 25 it's 3^7 * 2 * 2
You're correct Captainblack, the answer is 972.
I had a hard time understanding the inequality, but I guess as long as I take it for granted I should be fine.
Also, is it an actual fact that 3 is the best integer and is the best number to use for something like this?
number of terms in the sum and product, and that lots of 3's and a few 2's
will get you close to the maximum for the product, but its not definitive.
For instance you can get as large a product with 6 terms and with 7 when
the sum is 19.