1. ## Greatest Product

I'm working on this problem. I have an arbitrary number n. Then I'm finding out which will give me the biggest product when the numbers that make up the sum of n are multiplied together. The answer always seems to be a combination of 2's and 3's, but why is this?

2. suppose for the integer n you have $\sum_{i=1}^k a_i = n , a_i>0$. if one of them is larger than 3, lets say $a_1 \geqslant 4$ then $a_1 = 2 + (a_1-2)$ where $a_1 - 2 \geqslant 2$ so $2*(a_1 - 2) = (a_1 - 2) +(a_1 - 2) \geqslant a_1 - 2 + 2 = a_1$ and finally you get
$2 + (a_1 - 2) + \sum_{i=2}^k a_i = n$ and $2*(a_1-2)*\prod_{i=2}^k a_i \geqslant \prod_{i=1}^k a_i$
(notice that the inequality is strict if the integer is greater than 4)
so unless you only have 2's and 3's you can always get a better product

you can show in a similar way that you should prefer 3's over 2's
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