Determine $61! \pmod {71}$.
We start with Wilson's Theorem
$$ \begin{align} 70! & \equiv -1 \pmod {71} \\ \\ 70 \cdot 69 \cdot 68 \cdot 67 \cdot 66 \cdot 65 \cdot 64 \cdot 63 \cdot 62 \cdot 61! & \equiv -1 \cdot -2 \cdot -3 \cdot -4 \cdot -5 \cdot -6 \cdot -7 \cdot -8 \cdot -9 \cdot 61! \pmod {71} \\ \\ & \equiv -2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 61! \pmod {71} \\ \\ & \equiv -2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot (1) \cdot 61! \pmod {71} \\ \\ & \equiv -2 \cdot 4 \cdot (19) \cdot 7 \cdot 61! \pmod {71} \\ \\ & \equiv (-56) \cdot (19) \cdot 61! \pmod {71} \\ \\ & \equiv (1) \cdot 61! \pmod {71} \\ \\ & \equiv -1 \pmod {71} \\ \\ & \equiv 70 \pmod{71} \end{align} $$
This tells us $61! \equiv 70 \pmod {71}$.