Let \(S\) be the sum of the squares of the divisors of \(480\) (including itself). Compute the number of divisors of \(S\).
Solution
Answer: \(108\)
Since \(480=2^5\cdot3\cdot5\), expanding the following product gives the square of every divisor exactly once:
\[S=(1+2^2+2^4+2^6+2^8+2^{10})(1+3^2)(1+5^2).\]
Using the geometric series formula,
\[S=\frac{4^6-1}{3}\cdot10\cdot26=260\cdot65\cdot21=2^2\cdot3\cdot5^2\cdot7\cdot13^2.\]
The number of divisors is \((2+1)(1+1)(2+1)(1+1)(2+1)=108\).