Question

Allan writes down n positive integers so that the sum of those numbers is 23. When he multiples those numbers, he gets the maximum result. List all of the n positive integers that Allan has written.​

Answer 1

your answer should be 3, 3, 3, 3, 3, 3, 3, 2

Answer 2

• 3, 3, 3, 3, 3, 3, 3, 2

Explanation:

To get maximum product we need to optimize how we break down the numbers

Its obvious we don't want number 1 to be used as it won't change the product

Also we don't want to use big numbers, we can have four 5's instead of two 10's as it would give greater product, 5*5 > 10

What is best number?

• 2 and 3 are the smallest, 4 can be written as two 2's
• 5 = 2+ 3 and 2*3 = 6, so it is better to use 2's and 3's instead of 5
• 6 = 2+2+2, 2^3 = 8 and 6 = 3+ 3, 3^2 = 9, so it is better to use two 3's for 6
• 7 = 2+2+3, 2*2*3= 12, again 2's and 3's make greater product

So the best is to break the number down to 2's and 3's

• 23 = 2*10 + 3 ⇒ 2^10*3 = 3072
• 23 = 3*7 + 2 ⇒ 3^7*2 = 4374

As we see the maximum product we can get is 4374 with seven 3's and one 2.

Hence the numbers are:

• 3, 3, 3, 3, 3, 3, 3, 2