If the positive integer y is divisible by 3, 8, and 12, then which of the following must y be divisible by?

I. 24
II. 36
III. 48

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

I always search for the problems on GMAT club & then post my query. Sorry but I couldn't find anything similar to this & as Kaplan gave such a huge explanation I got a bit confused.

Oh ! Must & Could I will find it.

Thanks !!

Here is the detailed explanation:

Let’s look at the prime factors of 3, 8 and 12.

The prime factorization of 3 is 3.

Let's find the prime factorization of 8. We have 8 = 2 × 4 = 2 × 2 × 2.

Let's find the prime factorization of 12. We have 12 = 2 × 6 = 2 × 2 × 3.

Thus,


A multiple of 3, 8, and 12 must have at least three factors of 2 and one factor of 3, or 2 x 2 x 2 x 3.

Let's write the prime factorizations of 24, 36, and 48.


This is a Roman Numeral question, so let’s start with the numeral that appears most in the answer choices or Statement I. Since 24 has 3 factors of 2 and one factor of 3, y is a multiple of 24. So the correct answer will contain Statement I. We can eliminate choices (B) and (D) which do not contain Statement I.

Since 36 contains two factors of 3, but not three factors of 2, the correct answer will not contain II. We can eliminate choices (C) and (E) which contain Statement II. We have eliminated the 4 incorrect answer choices, so we know that choice (A) must be correct. Although we’ve eliminated all the wrong answer choices, we can that show that Statement III is incorrect. Since 48 contains four factors of 2, while y must contain at least three factors of 2, y does not have to be a multiple of 48, because y could contain exactly three factors of 2. The correct answer will not contain Statement III. Answer Choice (A) is correct.

The LCM of 3, 8, and 12 is 24. So y must be 24 or a multiple of of 24. Let’s test each answer choice.

We see that Roman numeral I is correct; if y = 24, then y/24 = 24/24 = 1.

Roman numeral II is not a multiple of 24, so it is not correct.

You might think that Roman numeral III, the value 48, should be correct, but it isn’t. If y were equal to 24, then y is NOT divisible by 48, because y/48 = 24/48 does not yield an integer as the answer.

The only answer that MUST be correct is y = 24.

Answer: A
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