Question

For any positive integer $x$, the $2$-height of $x$ is defined to be the greatest non-negative integer n such that $2^n$ is a factor of $x$. If $k$ and $m$ are positive integers, is the 2-height of $k$ greater than the 2-height of m?
(1) $k>m$
(2) $\frac{k}{m}$ is an even integer.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is B Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

1. Given that k > m, the 2-height of k can be greater than m (choose k = 4, which has a 2-height of 2, and choose m = 2, which has a 2-height of 1) and the 2-height of k can fail to be greater than m (choose k = 3, which has a 2- height of 0, and choose m = 2, which has a 2-height of 1); NOT sufficient.
2. Given that is an even integer, it follows that = 2n for some integer n, or k = 2mn. This implies that the 2-height of k is at least one more than the 2-height of m; SUFFICIENT.

• The correct answer is B; statement (2) alone is sufficient.