Divisibility & Primes Question

[sidney.bradley]

If 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

The correct answer was I & II. I'm confused! I did a factor tree of 4&6 and came up with 2,2,2,3. Since there is a 2 in both the factor tree of 4&6, I eliminated one of the 2s (which I felt like I learned....because you cant be sure whether the 2s are unique or shared). So then I concluded that xy had to be a multiple of something that can be made out of 2,2,3 which lead me to the answer of only II. Why is my logic off?

[navin018]

My obvious answer to the problem was 12. But, when I pulled out the first example of X and y, i.e., 4 and 6(x being multiple of 4 and y of 6), the number came up to 24(not 12, even though it is divisble by both 4 and 6). So, you may need to include that extra 2.

[RonPurewal]

Since there is a 2 in both the factor tree of 4&6, I eliminated one of the 2s

this is the problem right here.

you can't "eliminate" factors here, because you are actually multiplying two different numbers to produce a product, xy.
there are two 2's in "x", and one 2 in "y". since "x" and "y" are distinct entities, these are actually three different factors -- there's no "overlapping" here.

what you are trying to do here only works on repeated observations about the same number.
e.g., if i tell you that "x" is a multiple of 4, and then i tell you that "x" (the same "x") is a multiple of 6, then, yes, you only know that it's a multiple of 12.