Hello Stacy, :)

The number properties book has turned the light on divisibility. Outstanding! Can we discuss #23?

D23 OG answ: I & II

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
B. I & II
C. I & III
D. II & III
E. I, II, & III

Glad it's helping! This problem says X is a multiple of 4, which is the same as saying it is a multiple of 2*2. Y, then is a multiple of 6, or 2*3.

If we multiply X and Y, then we multiply 2*2*2*3. This represents, at the least, the factors of X and Y. (I say at the least because X, for example, could be 16, in which case X is a multiple of 2*2*2*2. But I don't know this for sure - I just know that X is a multiple of 2*2 at the least.) Finally, I need to notice that the questions is a "must" question. Sometimes they will ask questions like this with the word "must" and sometimes they will use the word "could" - which word is used can change the entire answer.

8 is a multiple of 2*2*2*3 because I can definitely construct 8 (or 2*2*2). I works.
12 is a multiple of 2*2*2*3 because I can definitely construct 12 (or 2*2*3). II works.
18 could be a multiple of XY but I can't say it definitely is because it is not a multiple of 2*2*2*3 (I would need 2*3*3 to construct 18). III could work but the problem specifies that it MUST work - so III doesn't work.