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.

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