Problem Solving

The LCM of three numbers is four times their GCF. Which of the following must be true of the numbers?

I. At least one of the numbers is odd.
II. Two of the three numbers must be the same.
III. At least one number is the same as GCF.

A. I only
B. III only
C. I and III only
D. I and II only
E. II and III only

[spoiler]OA=B[/spoiler]

Source: Veritas Prep

Gmat_mission wrote:The LCM of three numbers is four times their GCF. Which of the following must be true of the numbers?

I. At least one of the numbers is odd.
II. Two of the three numbers must be the same.
III. At least one number is the same as GCF.

A. I only
B. III only
C. I and III only
D. I and II only
E. II and III only

Try to prove that I, II and III DON'T have to be true.

To disprove I -- which states that at least one number must be odd -- let the GCF = 2.
Since the LCM = 4(GCF), the LCM = 8.
In this case:
The greatest factor common of all three numbers must be 2.
The least value divisible by all 3 numbers must be 8.
Thus, the three numbers could be 2, 4 and 8.

Since none of the numbers here are odd, eliminate any answer choice that includes I (A, C, and D).
Since all 3 numbers are different, eliminate any remaining answer choice that includes II (E).

The correct answer is B.