# The Question

This question is a maths challenge and well to explain it better I will use an example.
I have a number, say $20$
I can write this as $10 + 10$ (this is a make up of the number)
Now I multiply the two numbers to get $100 = 10 * 10$

The challenge is to make the highest possible product in this manner and find a rule for this.

Note :
The composition (make up) can consist of a number of numbers. hehe.
They have to be positive integers.

"Let n∈N be a positive integer. Let m∈N and a1,a2,…,am be positive integers such that a1+⋯+am=n. For what m and what a1,…,am is the product a1a2⋯am maximal" (Same, just rephrased, creds to SteamyRoot)

# A solution

Using the example $20$, to find the highest "production" number for this, we need to write out $20$ has a sum of as many as $3$ as we can use, so...
$20 = 3 + 3 + 3 + 3 + 3 + 3 + 2$
Now we multiply them together
$3*3*3*3*3*3*2 = 3^6 * 2 = 1458$

## marked as duplicate by TomGrubb, Ross Millikan, JimmyK4542, Nick Peterson, Community♦Dec 23 '16 at 18:02

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• Can we use Real numbers or only integers? – Tobi Alafin Dec 23 '16 at 17:49
• @TobiAlafin only integers – Max Dec 23 '16 at 17:51