Last Updated on May 10, 2023
GMAT OFFICIAL GUIDE DS
Solution:
We are given that for any positive integer x, the 2-height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. The “2-height of x” is a made-up term, but in math, we can make up any term as long as we provide a well-defined definition. So here, we are given that the 2-height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. If this needs clarification, here are some easy examples:
If x = 4, then the 2-height of 4 is 2 since 2^2 is a factor of 4 (but 2^3 isn’t).
If x = 5, then the 2-height of 5 is 0 since 2^0 is a factor of 5 (but 2^1 isn’t).
If x = 6, then the 2-height of 6 is 1 since 2^1 is a factor of 6 (but 2^2 isn’t).
Here, we want to determine if we have two positive integers k and m, whether the 2-height of k is greater than the 2-height of m.
Statement One Alone:
k > m
Statement one tells us that k is greater than m. As we can see from the three examples we’ve mentioned earlier, if k = 6 and m = 5, then the 2-height of k is greater than the 2-height of m. However, if k = 6 and m = 4, then the 2-height of k is NOT greater than the 2-height of m. Therefore, this is not enough information to determine whether the 2-height of k is greater than the 2-height of m.
Thus, statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.
Statement Two Alone:
m/k is an even integer.
When dividing two numbers that result in an even quotient, we must remember that we need to divide either an even by an even, or an even by an odd. In either case, we see that the 2-height of m will always be greater than the 2-height of k.
For instance, if we are dividing an even by an even to get an even quotient, m could be 4 while k could be 2 or m could be 12 and k could be 6. In either case, the 2-height of m is greater than the 2-height of k.
When dividing an even by an odd to get an even quotient, we see right away that an odd number will always have a 2-height of 0, and that an even number will, at least, have a 2-height of 1. Thus, the 2-height of m will always be greater than the 2-height of k.
Answer: B
