Let \(r\) and \(s\) be integers. Then \[\frac{6^{r+s}\times12^{r-s}}{8^r\times9^{r+2s}}\] is an integer if

\(r+s\le 0\),

\(s \le 0\),

\(r\le 0\),

\(r\ge s\).

Maybe we could break \(6, 12, 8\) and \(9\) down into their prime factors to rewrite the expression?