Need clarfication (Number properties - Div.and Primes)

Hello, BTG experts,

I just need clarification. I am a bit iffy about this problem:

Source: OG12 (Diagnostic Test)

23. If the positive integer x is a multiple of 4 and the positive integer y is a multiple of 6, then xy must be a multiple of which of the following?

I. 8
II. 12
III. 18

(A) II only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III

I answered (A) and got an incorrect answer. Here's my approach:
1. X is a multiple of 4. This means that the prime box of X contains {2,2,...} (notice that I used a partial prime box since X is a variable)
2. Y is a multiple of 6. This means that the prime box of Y contains {2,3,...} (also used a partial prime box here)

Now, here's the part where I think I got it wrong:
I got 12 as the LCM of xy by 2 x 2 x 3. I think this is wrong since we are dealing with two variables here. This means that there shouldn't be an "overlap" of 2.

(If we are just talking about just one variable here (say z), then there should be an overlap. In such a case, it's like looking at z at a different perspective.)

If I instead thought that there are two variables, then their LCM would be 2 x 2 x 2 x 3 = 24.

This makes 8 and 12 a multiple of xy; hence, the answer is (B) I and II only.

Is my thinking correct?

Thanks