melguy wrote:
What is the value of integer x?
1) The LCM of x and 45 is 225
2) The LCM of x and 20 is 300

We need to determine the value of x.

Statement One Alone:

The LCM of x and 45 is 225.

We can start by breaking 45 and 225 into prime factors.

45 = 3^2 x 5^1

225 = 3^2 x 5^2

In other words, 3^2 x 5^1 and some number x have an LCM of 3^2 x 5^2.

Thus, x could be the following:

5^2 = 25

5^2 x 3^1 = 75

5^2 x 3^2 = 225

Statement one is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

The LCM of x and 20 is 300.

We can start by breaking 20 and 300 into prime factors.

20 = 2^2 x 5^1

300 = 2^2 x 3^1 x 5^2

In other words, 2^2 x 5^1 and some number x have an LCM of 2^2 x 3^1 x 5^2.

Thus, x could be the following:

3^1 x 5^2 = 75

3^1 x 5^2 x 2^1 = 150

3^1 x 5^2 x 2^2 = 300

Statement two is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using statements one and two we see that the only value of x that fulfills both statements is 75.

Answer: C