For a positive number $n$, define $C(n)$ as the number of the integers $x$, for which $1 \lt x \lt n$ and
$x^3 \equiv 1 \bmod n$.
When $n=91$, there are $8$ possible values for $x$, namely: $9, 16, 22, 29, 53, 74, 79, 81$.
Thus, $C(91)=8$.
Find the sum of the positive numbers $n \le 10^{11}$ for which $C(n)=242$.
Solution
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