M04-15

Official Solution:


If \(p\) and \(q\) are nonzero integers and \(n=\frac{p}{q}\), is \(n\) an integer?

(1) \(n^2\) is an integer.

For \(n^2\) to be an integer, \(n\) must either be an integer or an irrational number such as \(\sqrt{3}\) (it's important to note that \(n\) cannot be a reduced fraction like \(\frac{2}{3}\) or \(\frac{11}{3}\) because in these cases \(n^2\) would not yield an integer). However, since \(n\) can be represented as the ratio of two integers (i.e., \(n=\frac{p}{q}\)), it cannot be an irrational number (by definition, an irrational number is any real number that cannot be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers). Consequently, the only remaining possibility is that \(n\) is an integer. This is sufficient.

(2) \(\frac{2n+4}{2}\) is an integer.

The above gives \(\frac{2n+4}{2}=n+2 =\text{integer}\). From this it follows that \(n\) must be an integer. Sufficient.


Answer: D