Practice!

Sample Problem

Several natural numbers add up to 23. What is the largest, possible product of these numbers?

Solution

In order to maximize the product, we need to write the sum in terms of only 2s and 3s. However, we want more 3s because 3 + 3 = 2 + 2 + 2, but 3 × 3>2 × 2 × 2. In other words, a 3 gives more for the dollar, so to speak. Therefore, if we write sum as

23 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 2,

we will attain the maximum product of

3 × 3 × 3 × 3 × 3 × 3 × 3 × 2 = 4374.