\(f(x)\) denotes the number of different prime factors of the integer \(x\), excluding \(1\) and \(x\) itself. For example, since \(12=2\times2\times3\), then \(f(12)=2\).
- Calculate \(f(121)\).
\(121=11\times 11\). So it has only one repeated prime factor, so \(f(121)=1\).
- Calculate \(f(17)\).
Since \(17\) is prime, that is, \(17=1\times 17\), and we are meant to exclude \(1\) and \(17\), we must have that \(f(17)=0\).
- Write down the smallest \(x\) for which \(f(x)=4\).
The smallest \(x\) for which \(f(x)=4\) will be the product of the four smallest prime numbers. So \(x=2\times3\times5\times7=210\).