Disprove the following statements:
(a) $a^2 ≡ b^2 \pmod n \implies a ≡ b \pmod n$
(b) $a × b ≡ 0 \pmod n \implies a ≡ 0 \lor b ≡ 0 \pmod n$
(c) $ac ≡ bc \pmod n \implies a ≡ b \pmod n$
We disprove the statements with counter-examples.
(a) Choosing $a=6, b=4, n= 10$ gives us
$ 6^2 = 36 \equiv 6 \pmod {10} $
$ 4^2 = 16 \equiv 6 \pmod {10} $
But $6 \equiv 4 \pmod {10}$ is false.
(b) Choosing $a=5, b=2, n=10$ gives us
$ 5 \times 2 = 10 \equiv 0 \pmod {10} $
But $2 \equiv 0 \pmod {10}$ and $5 \equiv 0 \pmod {10}$ are both false.
(c) Choosing $a=3, b=8, c=4, n=10$ gives us
$ ac = 3 \times 4 = 12 \equiv 2 \pmod {10} $
$ bc = 8 \times 4 = 32 \equiv 2 \pmod {10} $
But $3 \equiv 8 \pmod {10}$ is false.