M18-03

Official Solution:


If \(x\) and \(y\) are integers, is \(xy\) divisible by 3 ?

(1) \((x + y)^2\) is divisible by 9

Since \(x\) and \(y\) are integers, the given information implies that \(x + y\) is divisible by 3. However, this is not sufficient to determine whether \(xy\) is divisible by 3. For example, if \(x = 1\) and \(y = 2\), the answer is NO, but if \(x = 3\) and \(y = 3\), the answer is YES. This statement is not sufficient.

(2) \((x - y)^2\) is divisible by 9

Since \(x\) and \(y\) are integers, the given information implies that \(x - y\) is divisible by 3. However, this is not sufficient to determine whether \(xy\) is divisible by 3. For example, if \(x = 4\) and \(y = 1\), the answer is NO, but if \(x = 3\) and \(y = 3\), the answer is YES (remember that 0 is divisible by every integer, with the exception of 0 itself). This statement is not sufficient.

(1) + (2) If both \(x + y\) and \(x - y\) are divisible by 3, their sum \((x+y) + (x-y)=2x\) must also be divisible by 3. From the fact that \(2x\) is divisible by 3, we can conclude that \(x\) is divisible by 3. Since \(x\) and \(y\) are integers, it follows that \(xy\) must be divisible by 3. Sufficient.


Answer: C