Sunday, 4 January 2026

Exercise (4.2).4

Find $x$ where $x$ is the least non-negative residue such that

$$ x \equiv 8 × 9 × 10 × 11 × 16×17 × 18 × 19 \pmod {13} $$


We start as follows

$$ \begin{align} x & \equiv  8 \times 9 \times 10 \times 11 \times 16 \times 17 \times 18 \times 19 \pmod {13} \\ \\ & \equiv  (-5) \times (-4) \times (-3) \times (-2) \times 3 \times 4 \times 5 \times 6 \pmod {13} \\ \\ & \equiv  (-5) \times (-1) \times (-2) \times (-1) \times 5 \times 6 \pmod {13} \quad \quad \text{ using } 3 \times 4 \equiv -1  \\ \\ & \equiv  (-2) \times 6 \pmod {13} \quad \quad \text{ using } (-5) \times 5 \equiv 1 \\ \\ & \equiv -12 \pmod {13}\\ \\ & \equiv 1 \pmod {13} \end{align} $$

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