The sum of three distinct natural numbers is 20. What can be the maximum value of their product?

To find the maximum product of three distinct natural numbers that sum to 20, we need to consider the factors of 20.

Step 1: List down the possible sets of three distinct natural numbers that sum to 20.

Step 2: Calculate the product of the numbers in each set.

Step 3: Compare the products and find the maximum one.

However, there is a quicker way to solve this problem by using the arithmetic mean-geometric mean inequality (AM-GM inequality). The AM-GM inequality states that the arithmetic mean of a set of non-negative numbers is always greater than or equal to the geometric mean.

In this case, we want to maximize the product of three numbers (geometric mean) that sum to 20 (arithmetic mean). According to the AM-GM inequality, the product is maximized when the three numbers are as close to each other as possible.

So, we divide 20 by 3 to get approximately 6.67. Since we need distinct natural numbers, we choose 6, 7, and 7 as our three numbers. But since the numbers need to be distinct, we adjust them to 6, 7, and 8.

Step 4: Calculate the product of 6, 7, and 8.

6 * 7 * 8 = 336

So, the maximum product of three distinct natural numbers that sum to 20 is 336.