Problem Solving

NandishSS wrote:M is a positive integer less than 100. When m is raised to the third power, it becomes the square of another integer.
How many different values could m be?

A. 7
B. 9
C. 11
D. 13
E. 15

OA: B

We are given that m is a positive integer less than 100. We are also given that when m is raised to the third power, it becomes the square of another integer. In order for that to be true, m itself must (already) be a perfect square, since any perfect square raised to the third power will still be a perfect square, i.e., square of an integer. Thus we are looking for perfect squares that are less than 100. Since there are 9 perfect squares that are less than 100, namely, 1, 4, 9, ..., 64, and 81, the answer is 9.

Let's look at some examples to clarify this: Let's assume that m = 4 = 2^2. Now, let's raise m to the third power, obtaining m^3 = (2^2)^3 = 4^3 = 64, which is 8^2. Another illustration: let m = 25 = 5^2. Now, let's raise m to the third power, obtaining m^3 = (5^2)^3 = 25^3 = 15,625, which is 125^2.

(Note: By the way the problem is worded, "when m is raised to the third power, it becomes the square of another integer," 1 should not be counted as one of the 9 different values m could be, unlike all the other 8 values. For example, take the number 4: 4^3 = 64 = 8^2, which is the square of another integer, 8. However, 1^3 = 1 = 1^2, which is the square of the same integer. The correct way to word the problem is "when m is raised to the third power, it becomes the square of an integer.")

Answer: B