If x is a multiple of 18 and 60 GMAT Properties of Numbers

If x is a multiple of 18 and 60, then x must be divisible by which of the following?

I. 24

II. 36

III. 45

(A) I only

(B) II only

(C) I and II only

(D) II and III only

(E) I, II and III­

What we need to concern ourselves here is the meaning of the factors within 18 and 60 themselves. 

In short, we need to find the Least Common Multiple (LCM) of 18 and 60 and then test each of the statement numbers (24, 36, and 45) to see whether they also fit into this LCM.

First step: prime factorize each number.

This isn’t particularly hard to do, so I’ll take it for granted that you can figure that part out. 

The prime factorization of 18 is .

The prime factorization of 60 is

Second step: find the Least Common Multiple

That means that, if If x is a multiple of 18 and 60, the Least Common Multiple of these numbers (the smallest number divisible by each one) would be, effectively, the taking the highest power of each of the three present bases. 

That is, the in 18 has a higher power than the 3 in 60, so we keep it. Likewise, the in 60 has a higher power than the 2 in 18 so we keep that one. We also need to represent the 5, so that is kept. 

Therefore, the smallest number that would divide by both 18 and 60 can be defined as . 

This tracks, of course, if you think about it: 90 is a multiple of 18 but not of 60 because it doesn’t have two 2s in it. Therefore, 180 must be the smallest multiple that has the two 2s that are present in 60. 

So if 18 

Third step: factorize each of the statements

If x is a multiple of 18 and 60, I’m just saying this for SEO–bear with me.

Statement I: 

The prime factorization of 24 is . Note that our LCM does not include three 3s. Therefore, this number will create a fraction if we divide the LCM by it. 

That is, after everything is canceled out.  

This statement doesn’t work.

Statement II: 

The prime factorization of 36 is . This easily divides in. This statement works.

Statement III: 

The prime factorization of 45 is . This easily divides in. This statement works. 

Conclusion

And of course if If x is a multiple of 18 and 60, the answer is definitely (D) II and III only.