# Sum of GCDs

**Number Theory**Level 4

A positive integer \(n\) is called **ingenious** if
\[ \displaystyle\sum_{m=1}^{n} \gcd (m,n) \]
is a prime. Find the number of ingenious integers between \(3\) and \(100\) (inclusive).

**Details and assumptions**

You may refer to this list of primes.

As an explicit example, when \(n=3,\) we have \[\displaystyle\sum_{m=1}^{3} \gcd (m, 3) = \gcd (1, 3) + \gcd (2, 3) + \gcd (3, 3)= 5,\] which is a prime.

This problem was inspired by IMOSL 2004 N2.