Sunday, 4 January 2026

Exercise (4.2).5

Evaluate the least residue x such that

$$ x \equiv 2 (20!) \pmod {23} $$


Wilson's Theorem gives us a start

$$ \begin{align} 22! \equiv -1 \pmod {23} \\ \\ 22 \times 21 \times 20! \equiv -1 \pmod {23} \\ \\ (-1) \times (-2) \times 20! \equiv -1 \pmod {23} \\ \\ 2 \times 20! \equiv -1 \pmod {23} \end{align}$$

So the least residue is $x=-1$.


Note: the exercises asks for the least reside, not the least non-negative residue, so we interpret this as least magnitude.