Determine whether the following are true or false:
(a) $12 ≡ 232 \pmod{5}$
(b) $15 ≢ 5 \pmod{10}$
(c) $12 ≡ −1 \pmod {11}$
(d) $365 ≢ 1 \pmod {7}$
(e) $−65 ≡ −29 \pmod {12}$
(f) $−43 ≢ −46 \pmod {2}$
(a) We have
$$ 12 \equiv 2 \pmod{5} $$
$$ 232 \equiv 2 \pmod{5} $$
So it is true that
$$ 12 \equiv 232 \pmod{5} $$
(b) We have
$$ 15 \equiv 5 \pmod{10} $$
$$ 5 \equiv 5 \pmod{10} $$
So it is false that
$$ 15 \not \equiv 5 \pmod{10} $$
(c) We have, using the least non-negative residue
$$ 12 \equiv 1 \pmod {11} $$
$$ -1 \equiv 10 \pmod {11} $$
So it is false that
$$ 12 ≡ −1 \pmod {11} $$
(d) We have
$$ 365 \equiv 1 \pmod{7} $$
So it is false that
$$365 \not \equiv 1 \pmod {7}$$
(e) We have, using the least non-negative residue
$$ -65 \equiv 7 \pmod{12} $$
$$ -29 \equiv 7 \pmod{12} $$
So it is true that
$$−65 \equiv −29 \pmod {12}$$
(f) We have, using the least non-negative residue
$$ -43 \equiv 1 \pmod {2}$$
$$ -46 \equiv 0 \pmod {2}$$
So it is true that
$$−43 \not \equiv −46 \pmod {2}$$