Data Sufficiency

The least common multiple (LCM) of two integers is a short-hand way of telling you which prime factors they share, and which ones they don't share. For example, if the LCM of x and y is 70, you know that between them they have a 2, a 5, and a 7. You also know that neither of them has a factor of 3, or two factors of 5, etc.

This question is asking about the LCM of 6, 9 and x. In other words, it's asking what prime factors x has that 6 and 9 don't. You could start by finding the LCM of 6 and 9. You can probably figure out pretty quickly that the LCM is 18, but if you wanted to prove it... find the prime factors of each: 6 = 3*2, 9 = 3*3. Then multiply all the factors, except any that overlap. LCM = 2*3*3 = 18. This is another way of saying that the combined factors of 6 and 9 are 2, 3 and 3.

The LCM of x, 6, and 9 will be the same as the LCM of x and 18. So, we could rephrase our question as - what prime factors does x have that 18 doesn't? Or to put it another way, what prime factors does x have other than a 2, 3 and 3?

(1) The least common multiple of x and 6 is 30.

If the LCM of x and 6 is 30, that means that the combined factors of x and 6 are 2, 3, and 5. So, x has a 5 in it that 6 doesn't, but no other primes that 6 doesn't. (It may or may not have a 2 or a 3, but that doesn't matter. We don't have to know the value of x, we just need to know the prime factors that it doesn't share).

If the combined factors of x and 6 are 2, 3 and 5, then the combined factors of x, 6 and 9 are 2, 3, 3, and 5, so we know the LCM is 90. SUFFICIENT

(2) The least common multiple of x and 9 is 45.

Same logic applies here. If the LCM of x and 9 is 45, then the combined factors of x and 9 are 3, 3, and 5. X has a 5 in it, but no other prime factors that 9 doesn't have. So, the combined prime factors of 6, 9 and x must be 2, 3, 3, and 5, and the LCM is 90. SUFFICIENT

We didn't need to test numbers and check here (although that technique would also work). It's quicker to ask the conceptual question - what does the LCM tell me about the prime factors that it has, and the prime factors it doesn't have?

Answer: D