M05-36

Official Solution:


If \(a\), \(b\), and \(c\) are distinct positive integers, is \(\frac{(\frac{a}{b})}{c}\) an integer?

Firstly, simplify the expression \(\frac{(\frac{a}{b})}{c}\) to \(\frac{a}{bc}\).

(1) \(\frac{a}{c} = 3\).

Substitute the value of \(\frac{a}{c}\) into the simplified expression to get \(\frac{a}{bc}=\frac{3}{b}\). If \(b\) equals 1 or 3, then the result is an integer. However, if \(b\) represents any other positive integer, the outcome is not an integer. Not sufficient.

(2) \(a = b + c\).

Substituting \(a\) into the equation gives \(\frac{a}{bc}=\frac{b+c}{bc}=\frac{b}{bc}+\frac{c}{bc}=\frac{1}{c}+\frac{1}{b}\). For this expression to become an integer, either \(b=c=1\) or \(b=c=2\) would need to be true. However, we know that \(b\) and \(c\) are distinct positive integers, so neither case is possible. Therefore, \(\frac{1}{c}+\frac{1}{b}\) is not an integer. Sufficient.


Answer: B