What is the greatest integer factor of x if both x and y are integers?

Veritas official Solution
In this question, you are asked for the greatest integer factor of x
- the greatest integer that will divide evenly into x
- given that x
and y
are integers.

Statement (1) gives you that the least common multiple of x
and y
is 12. Remember that you can find the least common multiple one of two ways - either by listing out multiples of both numbers until you find a match or by combining the prime factorizations of both numbers. Since you don't have any values for x
or y
, listing out multiples is going to be impractical, so your next step should be to find the prime factorization of 12.

12 can be rewritten as (4)(3)
, or (22)(3)
.

However, notice that x
could be almost any combination of these numbers based on what you know so far -- it could be 12, 3, 2, or 4 -- and because of that it could have multiple greatest integer factors. Statement (1) is not sufficient. Eliminate (A) and (D).

Statement (2) gives you that the greatest common factor of x
and y
is 6. Remember that the greatest common factor is going to be the largest number that divides into both x
and y
respectively. However, that means that - as long as x
is a multiple of 6, that x
could be anything. Again, by itself statement (2) is insufficient. Eliminate (B).

Taken together, you know that x
must be a multiple of 6 between 6 and 12. However, since x
could be 12, there is no way of determining the value of x
- and therefore no way of determining the largest factor of x
. The statements are therefore insufficient even when taken together and answer choice (E) is correct.

I don't understand the answers with this one.

It says he least common multiple of x and y is 12. Therefore potential values for X and Y are;
1+12
12+1
6+2
2+6
4+3
3+4

The greatest common factor of x and y is 6
6+12
12+6

Am I missing something here? The pairing of y and x being 6 and 12 is the only possibility for the 2nd statement but this is not an option in the first statement.

Hmm I picked A. The original condition states that it is looking for the greatest integer factor. Under Statement 1, the combination in which X could be the highest would be when X = LCM = 12. How is this not sufficient? I understand that there are multiple combinations of factors with LCM = 12, but the highest possible value for X out of any of the combinations will always be 12.