If a positive integer q is divisible by both 3 and 11, then q must als

Question

If a positive integer q is divisible by both 3 and 11, then q must also be divisible by which of the following?

I. 14
 II. 33
 III. 66

A. I only
B. II only
C. III only
D. I and II
E. II and III

Hello, I have been using The Princeton Review's Math Workout for the Gmat, 3rd Edition, and encountered a question that I don't fully understand.

Practice Set, pg. 58, #5

I chose E, but the correct answer is B. The explaination given is as follows:

Numbers that are divisble by 3 and 11 include 33, 66, 99, 132, and so forth. You can eliminate I and III because not all these numbers are divisible by 14 or 66. All these numbers are divisble by 33, so choose B.

I still don't understand why 66 is eliminated. I would appreciate it if someone could explain this to me; I am ridiculously slow-witted with math, but what can ya do?

Thanks!

dreambeliever · Answer

66 is eliminated because 33 is not divisible by 66. it is just divisible by 3 and 11.

mijou · Answer

But what if q is 66? if q is 66 its divisible by 3, 11, 33 and itself 66. No?

IanStewart · Answer

This hinges on the wording of the question. The question asks what q *must* be divisible by. It certainly is *possible* that q=66, say, and that q is divisible by 66, but it's also possible that q = 33, in which case q is not divisible by 66. So q does not *need* to be divisible by 66 here, which is why we can rule out III. I'd add that it's also *possible* for q to be divisible by 14 (it might be that q = 3*11*14 = 462, for example) but q certainly does not need to be divisible by 14. However, if q is divisible by 3 and 11, then q will always be divisible by the LCM of 3 and 11, which is 33, so q must be divisible by 33.

mijou · Answer

Well fancy that, thanks a lot IanStewart! I get it!=D

EMPOWERgmatRichC · Answer

Hi All,

We're told that Q is a positive integer that is divisible by BOTH 3 and 11. We're asked which of the following MUST also divisible evenly into Q. This question can be approached in a couple of different ways, including by TESTing VALUES.

With these types of prompts, it often helps to choose the SMALLEST value that fits everything that you're told. Here, the smallest positive value of Q that is divisible by 3 and 11 is 33.

IF.... Q=33...

I. 14 
14 does NOT divide evenly into 33. 
Eliminate Answers A and D

II. 33 
33 DOES divide evenly into 33, so we'll keep this option for now.

III. 66 
66 does NOT divide evenly into 33. 
Eliminate Answers C and E.

There's only one answer remaining...

Final Answer:  B

GMAT assassins aren't born, they're made, 
Rich

ScottTargetTestPrep · Answer

Since q is divisible by both 3 and 11, it’s also divisible by the LCM of 3 and 11 (and all the factors of this LCM). Since the LCM of 3 and 11 is 3 x 11 = 33, q is divisible by 33. (Note: q is not divisible by 14 and 66 since they are not factors of 33.)

Answer: B

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