In this question, we are given that 72 can be expressed as the product of two positive integers, m and n, where m is greater than 1 and less than n. We need to determine how many possible values of m are there.

To solve this problem, we can start by finding the factors of 72. The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

Now, we need to eliminate the factors that do not meet the condition of m being greater than 1 and less than n. Since m cannot be equal to 1, we eliminate 1 from the list of factors.

If we consider the other factors one by one, we can see that for each factor, there is exactly one corresponding factor that satisfies the condition of m being less than n. For example, if we take 2 as a factor, we know that m can be 2 and n can be 36. Similarly, if we take 3 as a factor, m can be 3 and n can be 24.

Therefore, there is only one possible value of m that satisfies the given conditions.