Monday, 17 November 2025

Exercise (3.2).3

Give three different examples which satisfy the following:

$a × b ≡ 0 \pmod n$ but $a ≢ 0 \pmod n$ and $b ≢ 0 \pmod n$.


Example 1

$a = 2, b = 3, n = 6$ gives $2 \times 3 \equiv 0 \pmod 6$ but $2 \not \equiv 0 \pmod 6$ and $3 \not \equiv 0 \pmod 6$.


Example 2

$a = 2, b = 5, n = 10$ gives $2 \times 5 \equiv 0 \pmod {10}$ but $2 \not \equiv 0 \pmod {10}$ and $5 \not \equiv 0 \pmod {10}$.


Example 3

$a = 3, b = 5, n = 15$ gives $3 \times 5 \equiv 0 \pmod {15}$ but $3 \not \equiv 0 \pmod {15}$ and $5 \not \equiv 0 \pmod {15}$.