M16-03

Official Solution:


Is \(x^2 y^4\) an integer divisible by 9 ?

The crucial observation to solve this question is to note that we are not given that \(x\) and \(y\) are integers.

(1) \(x\) is an integer divisible by 3

If \(x = 3\) and \(y=any \ integer\), then the answer would be YES. However, if \(x=3\) and \(y=\frac{1}{2}\), then the answer would be NO. Not sufficient.

(2) \(xy\) is an integer divisible by 9

If \(x = 9\) and \(y=any \ integer\), then the answer would be YES. However, if \(x=6\) and \(y=\frac{3}{2}\), then the answer would be NO. Not sufficient.

(1)+(2) We can use the same example as in the second statement:

If \(x = 9\) and \(y=any \ integer\), then the answer would be YES. However, if \(x=6\) and \(y=\frac{3}{2}\), then the answer would be NO. Not sufficient. Essentially even when taken together the statements are not sufficient to deduce that \(y\) is an integer, which would guarantee that \(x^2 y^4\) is divisible by 9. Nor can we deduce that \(y\) is such a noninteger, for which \(x^2 y^4\) is divisible by 9 (for example, \(x = 3^3\) and \(y=\frac{1}{3}\)).


Answer: E