Compute \(d(30375)\).

Compute \(\sigma(1440)\).

Compute \(\phi(4050)\).

Find the smallest positive integer that has exactly \(30\) positive divisors.

Determine the least residue of \(163^{192}\) modulo \(360\).

Prove that \(\phi(n)=n/2\) if and only if \(n=2^k\) for some positive integer \(k\).

Show that if \((m,n)=1\) and \(n\) is abundant, then \(mn\) is abundant.

Let \(n=dm\). Show that there are \(\phi(m)\) positive integers less than \(n\) whose greatest common divisor with \(n\) is \(d\).