Basic properties of congruences (i) We have

a ≡ a (mod m)

for each a. (ii) If

a ≡ b (mod m)

then

b ≡ a (mod m).

(iii) If

a ≡ b (mod m) and b ≡ c (mod m) then

a ≡ c (mod m).

(iv) If

a ≡ b (mod m) and c ≡ d (mod m) then

a + c ≡ b + d (mod m),
a − c ≡ b − d (mod m)

and

ac ≡ bd (mod m).