M31-53

Official Solution:


If \(a\) and \(b\) are non-negative integers, is \(a > b\)?

(1) \(6^a = 36^b\).

Simplify: \(6^a = 6^{2b}\). Since the bases are equal, we can equate the powers: \(a=2b\). If \(a=b=0\), then \(a\) is NOT greater than \(b\). However, if \(a=2\) and \(b=1\), then \(a\) IS greater than \(b\). This information is not sufficient to determine whether \(a > b\).

(2) \(5^a = 35^b\).

If both \(a\) and \(b\) are positive integers, then \(5^a\) would equal some multiple of 7 (because 35 = 5 * 7). However, this is not possible since raising 5 to any positive integer power results in a number with only 5 as its prime factor. Consequently, both \(a\) and \(b\) must be 0, which means \(a\) is NOT greater than \(b\). This information is sufficient to answer the question.


Answer: B