stonecold wrote:

If k is an even integer, what is the smallest possible value of k such that 3675*k is the square of an integer?

A.3

B.9

C.12

D.15

E.20

If 3675*k is a perfect square, then its prime factorization must contain only even exponents. Let’s begin by prime factoring 3,675.

3675 = 25 x 147 = 5 x 5 x 49 x 3 = 5 x 5 x 7 x 7 x 3 = 5^2 x 7^2 x 3^1

We can see that 3,675 is not a perfect square because its prime factorization contains an odd exponent (that is, 3^1). If k only has to be an integer, then the smallest value k can be is 3, since 3675*k would be 5^2 x 7^2 x 3^2, a perfect square. However, since the requirement is that k must be an even integer, we need k to be divisible by an even perfect square (notice that 3675 doesn’t have any even prime factors). Since the smallest even perfect square is 2^2 = 4, the smallest possible value of k is 3 x 4 = 12, so that 3,675*k is a perfect square.

In fact, if k = 12, then 3,675*k = 5^2 x 7^2 x 3^2 x 2^2, which is a perfect square.

[Note: The smallest possible value of k such that 3675*k is the square of an integer is actually 0, since 3675*0 = 0, which is 0^2. To avoid this case, the problem should be stated as “If k is a positive even integer…such that 3675*k is the square of a positive integer?”]

Answer: C