# Given a number N, how to construct a set of different numbers that has a maximal product, and the sum of these numbers equal N?

Note that:

1. N is positive integer.
2. The set also consists of positive integers.
3. The set consists of different integers. (The thread suggested by @hardmath doesn't have this constraint.)

For example:

if $N = 4$, we can construct the set as ${4}$, and the product is 4.

if $N = 5$, we can construct the set of two elements ${2, 3}$, and their product is 6.

if $N = 7$, we can construct the set of two elements ${3, 4}$, and their product is 12.

if $N = 31$, the set is {2, 3, 5, 6, 7, 8}.

How to solve this problem for a general $N$? And how can you prove the correctness of your solution?