M05-16

Official Solution:


If \(x\) and \(y\) are positive integers and \(xy\) is divisible by prime number \(p\). Is \(p\) an even number?

Note that since \(p\) is a prime number and the only even prime number is 2, the question is essentially asking whether \(p=2\).

(1) \(x^2 * y^2\) is an even number.

Having \(x^2 * y^2\) as even implies that \(xy\) is even, which means that at least one of the variables, \(x\) or \(y\), is even. However, knowing that some even number is divisible by the prime number \(p\) is not sufficient to determine whether \(p=2\). For instance, if \(xy=6\), then \(p\) could be either 2 or 3.

(2) \(xp = 6\).

Given that \(x\) is a positive integer and \(p\) is a prime number, either \(x=2\) and \(p=3\) (answer NO) or \(x=3\) and \(p=2\) (answer YES). This information alone is not sufficient to determine whether \(p\) is an even number.

(1)+(2) If \(y=6\), then \(xy\) is even, which satisfies the first statement regardless of the value of \(x\). Therefore, we have no constraints on the value of \(x\). Based on statement (2), \(x\) can take either of the two values, 2 or 3, which means that \(p\) can also take either of the two values, 2 or 3, respectively. Combining the statements is not sufficient to determine whether \(p\) is an even number.


Answer: E