# Find a set of numbers that add to 25 and can multiply to the highest product.

I left class today with a riddle. Find a set of numbers (does not specify what type) that add to 25, but must multiply to a number as high as possible. For example, 10+10+5 = 25, 10*10*5 = 500.

I started with 5+5+5+5+5 (5^5 = 3125) but I realized I can break up these numbers more. It ended up with 2.7182 adding to itself 9.1969 times, which resulted in a product of 9866.4344

I also realized that it should be a number added and multiplied by itself. What is this number? Or is it a set of different numbers?

• "Find a set of numbers" - by definition, set cannot contain multiple objects, how is then $(10,10,5)$ a valid example? – user499203 Nov 24 '17 at 3:24
• Unlike ThePirateBay, I'm not worried about having a number appear twice, but a set definitely can't contain a non-integer number of elements! – Cheerful Parsnip Nov 24 '17 at 3:32
• New to Stack, let's just assume I'm a five-year-old. A set of numbers is just a list of random numbers - yes you can repeat numbers. – Cillowing Nov 24 '17 at 3:34

From the question I suppose you're allowed to use non-integer numbers which also may appear arbitrary number of times. Then the question really makes no sense, because such list of numbers doesn't exist. It doesn't exist because (if we assume that numbers in the list are finite) we can always find another list whose sum is $25$, but whose product is bigger than the given list.
For example, let $n\to\infty$ be a real number, then the list $$(2n,-n,-n,25)$$ sums up to $25$, but it's product tends to infinity.