Find the sum of all positive integers $n$ for which \[ \frac{15 \cdot n !^{2}+1}{2 n-3} \] is an integer.
Solution: It is clear that $n=1$ and $n=2$ work so assume that $n>2$. If $2 n-3$ is composite then its smallest prime factor is at most $\frac{2 n-3}{2}<n$ so will be coprime to $15 \cdot n !^{2}+1$. Therefore assume that $2 n-3=p$ is prime. We can rewrite the numerator as \[ (-1)^{n} \cdot 15 \cdot\left(1 \cdot 2 \cdots \frac{p+3}{2}\right) \cdot\left(\frac{p-3}{2} \cdot \frac{p-1}{2} \cdots(p-1)\right)+1 \quad(\bmod p) \] By Wilson's Theorem, $(p-1) ! \equiv-1(\bmod p)$, so the expression simplifies to \[ (-1)^{n+1} \cdot 15 \cdot \frac{p-3}{2} \cdot \frac{p-1}{2} \cdot \frac{p+1}{2} \cdot \frac{p+3}{2}+1 \equiv(-1)^{n+1} \cdot \frac{135}{16}+1 \quad(\bmod p) \] If $p \equiv 3(\bmod 4)$, then we have \[ \frac{135+16}{16} \equiv \frac{151}{16} \equiv 0 \quad(\bmod p) \] If $p \equiv 1(\bmod 4)$, then we have \[ \frac{135-16}{16} \equiv \frac{119}{16} \equiv 0 \quad(\bmod p) \] So $p$ must be a prime divisor of 151 or 119 , which means that $p \in\{7,17,151\}$. All of these numbers work aside from $7($ because $7 \equiv 3(\bmod 4))$ and the corresponding values of $n$ are 10 and 77 . The sum of the solutions is then $1+2+10+77=90$. $\fbox{90}$.