Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 4 / 1E / a / Solution

Since $31$ is a prime number, the Wilson theorem states that $30!\equiv -1(\mathrm{mod}31)$. But $30!\equiv (-1)(-2)28!=2\cdot 28!$, so the modular inverse of $2$, which is $16$, givesThe Fermat little theorem states that $a^{p-1}\equiv 1(\mathrm{mod}p)$ when $p$ is prime and $p\nmid a$. Therefore $13^{30}\equiv 1(\mathrm{mod}31)$, andsince $14\cdot 20=280=9\cdot 31+1$. Combining these modular congruences,This representative lies between $0$ and $30$, so it is the least nonnegative residue.