Monday, 17 November 2025

Exercise (3.2).5

Give three different examples which satisfy

$a × b ≡ 0 \pmod p ⇒ a ≡ b ≡ 0 \pmod p$

where $p$ is prime.


Example 1

$a = 3, b = 3, p = 3$ gives $3 \times 3 \equiv 0 \pmod 3$, and $3 \equiv 3 \equiv 0 pmod 3$.


Example 2

$a = 2, b = 4, p = 2$ gives $2 \times 4 \equiv 0 \pmod 2$, and $2 \equiv 4 \equiv 0 pmod 2$.


Example 3

$a = 25, b = 5, p = 5$ gives $25 \times 5 \equiv 0 \pmod 5$, and $25 \equiv 5 \equiv 0 pmod 5$.