Factors and Multiples in GMAT Data Sufficiency

Is the positive integer n a multiple of 40?

1)  20 is a factor of n^2
2)  (n^3)/128 is an integer.

I like to think of these “is X a multiple of Y?” questions as: “does X contain ALL of Y’s factors?”

40 = 2 x 2 x 2 x 5, or three 2’s and one 5.

This is a Yes or No question.

If Yes, then “n” will have three 2’s and one 5 as factors.
If No, then “n” will be account for all factors.

We don’t care whether the answer is actually “yes” or “no” (no horse in this race! :)), but we just need it to be 100% certain.

1) 20 is a factor of n^2

Okay, so n^2 = n x n, and they are saying that n x n will have all the factors of 20. 20 = 2 x 2 x 5

So basically 2 x 2 x 5 would evenly divide into n x n. But hang on a minute, we have two n’s in that hypothetical numerator, and the factors don’t evenly split up!

This tells us that “n” must have AT LEAST one 2 and one 5 as factors. So, it’s possible the answer is YES, if n also has two more 2’s, but what if n = 10? Then we’d get a NO answer. This is insufficient. Cross off answer choices A and D.

Let’s apply the same logic to Statement 2:

2) n^3/128 is an integer.

128 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2, since we’re splitting these eight 2’s amongst three n’s, it must be that we’re missing a 2, and that each “n” has AT LEAST three 2’s in it. What we don’t know anything about is whether or not it has a 5. If it does, the answer is YES. If it doesn’t, the answer is NO. This is insufficient; cross off answer choice B.

Combined:

“n” must have one 5 as per Statement 1, and “n” must have three 2’s as per Statement 2. Therefore the answer will always be YES, that n is a multiple of 40. (Also, remember that every number is factor and a multiple of itself. So let’s say n = 40. 40 is a multiple of 40, so that would give us a YES answer.)

The answer is (C).

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