M02-18

Official Solution:


If \(m\) and \(n\) are positive integers, and \(x = 2^m3^n\), is \(m < n\) ?

(1) \(x\) is divisible by 144.

Factoring 144, we get \(144 = 2^4 * 3^2\).

Hence, this statement implies that \(m ≥ 4\) and \(n ≥ 2\). However, without knowing the upper limits of \(m\) and \(n\), we cannot determine whether \(m < n\). Not sufficient.

(2) \(x\) is not divisible by 648.

Factoring 648, we get \(648 = 2^3 * 3^4\).

Hence, this statement implies either \(m < 3\) or \(n < 4\), or both. For instance, if \(m = 2\) and \(n = 10\), then \(m < n\). But if \(m = 10\) and \(n = 3\), then \(m > n\). Not sufficient.

(1)+(2) From (1) we know \(m ≥ 4\), so the condition \(m < 3\) from (2) cannot be true. Therefore, based on (2), \(n < 4\) must be true. This means \(m ≥ 4\) and \(n < 4\), leading to the conclusion that \(m > n\). Sufficient.


Answer: C