Check whether the following congruences satisfy the rule,
$$ac ≡ bc \pmod n \implies a ≡ b \pmod n$$
(a) $5 × 4 ≡ 5 × 7 \pmod 3$
(b) $9 × 12 ≡ 9 × 8 \pmod 6$
(c) $6 × 11 ≡ 6 × 7 \pmod 8$
(d) $13 × 21 ≡ 13 × 7 \pmod {26}$
(e) $13 × 31 ≡ 13 × 5 \pmod {26}$
(f) $101 × 35 ≡ 101 × 66 \pmod {31}$
(a) Here $a=4, b=7$ and $4 \equiv 7 \pmod 3$ is true.
This is because $4-7 = -3 = (-1) \times 3$.
(b) Here $a=12, b=8$, but $12 \equiv 8 \pmod 6$ is false.
This is because $12 \equiv 0 \pmod 6$.
(c) Here $a=11, b=7$, but $11 \equiv 7 \pmod 8$ is false.
This is because $11 \equiv 3 \pmod 8$.
(d) Here $a=21, b=7$, but $21 \equiv 7 \pmod 26$ is false.
This is because $21 \equiv 21 \pmod 26$.
(e) Here $a=31, b=5$, and $31 \equiv 5 \pmod 26$ is true.
This is because $31 -5 = 26 = (1) \times 26$.
(f) Here $a=35, b=66$, and $35 \equiv 66 \pmod 31$ is true.
This is because $35 -66 = -31 = (1) \times 31$.