Data Sufficiency

If x is a positive integer, what is the least common multiple of x, 6, and 9 ?
(1) The least common multiple of x and 6 is 30.
(2) The least common multiple of x and 9 is 45.

Target question: What is the LCM of x, 6 and 9?

I'll show you two different approaches.
This first approach uses requires us to be able to think of pairs of values that have given LCM's.
This is a useful skill to have on the GMAT.

Statement 1: The least common multiple of x and 6 is 30.
So, what are some possible values of x?
If the LCM of x and 6 is 30, then x could equal 5, 10, 15 or 30
Let's check each possible value of x.
- If x = 5, then the LCM of x, 6, and 9 is 90
- If x = 10, then the LCM of x, 6, and 9 is 90
- If x = 15, then the LCM of x, 6, and 9 is 90
- If x = 30, then the LCM of x, 6, and 9 is 90
So, even though x can have several different values, it must be the case that the LCM of x, 6, and 9 is 90
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The least common multiple of x and 9 is 45.
So, what are some possible values of x?
If the LCM of x and 6 is 30, then x could equal 5, 15 or 45
Let's check each possible value of x.
- If x = 5, then the LCM of x, 6, and 9 is 90
- If x = 15, then the LCM of x, 6, and 9 is 90
- If x = 45, then the LCM of x, 6, and 9 is 90
So, even though x can have several different values, it must be the case that the LCM of x, 6, and 9 is 90
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent

If x is a positive integer, what is the least common multiple of x, 6, and 9 ?
(1) The least common multiple of x and 6 is 30.
(2) The least common multiple of x and 9 is 45.

APPROACH #2

Target question: What is the LCM of x, 6 and 9?

ASIDE: The LCM tells us about the prime factors that numbers have in common.
For example: The LCM of 20 and 12 is 60
60 = (2)(2)(3)(5). So, the prime factorization of 60 has two 2's, one 3, and one 5.

Now examine the prime factorizations of 20 and 12
20 = (2)(2)(5)
12 = (2)(2)(3)
Notice that each prime factorization has no more than two 2's, one 3, and one 5 in it.
Also notice that the combined prime factorizations of 20 and 12 account for the two 2's, one 3, and one 5 that we find in the prime factorization of 60.

Statement 1: The least common multiple of x and 6 is 30
30 = (2)(3)(5)
6 = (2)(3), so we've already accounted for the one 2 and one 3 in the prime factorization of 30
We're missing only a 5
So, the prime factorization of x must have a 5 in it.
The prime factorization of x could also have a 2 or 3 in it, but they aren't required.
So, the possible values of x are 5, 10 (aka 5 times 2), 15 (aka 5 times 3) and 30 (aka 5 times 2 times 3)
As we saw in my earlier post, if we consider all of these possible values of x, the LCM of x, 6 and 9 is always 90
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The least common multiple of x and 9 is 45.
45 = (3)(3)(5)
9 = (3)(3), so we've already accounted for the two 3's in the prime factorization of 45
We're missing only a 5
Using the same logic as above, the possible values of x are 5, 15 and 45
If we consider all of these possible values of x, the LCM of x, 6 and 9 is always 90
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent