If x is divisible by 12 and 18, which of the following must divide

It depends on the wording.

If something has to divide into x, when x is 36, then the answer can only be I and II.

If x is divisible by 12 and 18, which of the following must divide evenly into x?

Grammaticaly speaking, when you say dividing two into four, the answer is two
In the question stem it says divide evenly into x, so that would mean 6 and 36.
In this wording, it says that k must divide evenly into x, leaving no remainder. In other words, k is a divisor of X.This is also the case in the princeton review books.

Answer is B

Answer B
First number divisible by both 12 and 18=36. Now, 72 does not evenly divide into 36. Therefore, only 6 and 36 satisfy

If X can be divided evenly by 12 and 18 , it must be divided by the LCM of the 2 numbers
LCM of 12 and 18 = 36

So if hypothetically a number can be divided by 12, 18 and 36 , it will also be divided by 6.

Lets check the third number 72.

What if number to be divided turns out to be 36. Can it be divided by 72 ? NO

Hence, Correct Option is B

If x is divisible by 12 and 18, which of the following must divide evenly into x?

I. 6
II. 36
III. 72

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

12 = 2^2 x 3 and 18 = 2 x 3^2. If a number is divisible by both 12 and 18, the number has at least two 2's and two 3's in its prime factorization.

So we can write x = 2^2 x 3^2 x m (m is an integer) or x = 36m

This implies x is always divisible by 6. (2 x 3). I is always true. Eliminate C and D.
II is always true as well. As x=36m; x is always divisible by 36. Eliminate A.

III not always true. Now x=36m. if m=1; x=36. this is not divisible by 72. Hence not always true.

Eliminate E.

this leaves B.

General expression for x=> 36*p for some integer p
so C is the answer as 36 and 72 are possible but 6 is not.

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