Need clarfication (Number properties - Div.and Primes)

[mdavidm_531]

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

[Bill@VeritasPrep]

I represented x and y differently.

X is a multiple of 4, so we know it is the product of 4 and an integer (that integer could very well be 1). Thus, I said that x=4i.

Y is a multiple of 6, so we know it is the product of 6 and an integer. I said that y=6j

From there, we know that xy = 4i*6j = 24ij. Since we know 24 is a factor of xy, any factors of 24 are also factors of xy. This includes both 8 and 12, but it does not include 18.

[mdavidm_531]

I represented x and y differently.

X is a multiple of 4, so we know it is the product of 4 and an integer (that integer could very well be 1). Thus, I said that x=4i.

Y is a multiple of 6, so we know it is the product of 6 and an integer. I said that y=6j

From there, we know that xy = 4i*6j = 24ij. Since we know 24 is a factor of xy, any factors of 24 are also factors of xy. This includes both 8 and 12, but it does not include 18.

Thanks, Bill,

Anyway, let's see if I get this right.

Source: OG12 DS #82

If positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105?
x = {2,3,..} y = {2,7,..} which means xy = {2,2,3,7,..} <-- is this correct? two 2's?

105 = {3,5,7}

Rephrase: There's already a 3 and a 7 in the prime box of xy, we need a 5.

(1) x is a multiple of 9 x = {2,3,3,...} - Insufficient
(2) y is a multiple of 25 y = {5,5,...} - Sufficient

Answer: B

[Bill@VeritasPrep]

Yep, you got it!