Let a, b, c, and d be integers, and let n be a positive integer. Prove that if a is congruent to c mod n and b is congruent to d mod n, then (a-b) is congruent to (c-d) mod n

Answer 1

Given:
a≡c mod n
b≡d mod n
Prove that (a-c)≡(b-d) mod n.

Let
a=kn+r ... k,r ∈ ℤ⁺
c=ln+r ... l,r ∈ ℤ⁺
Subtract:
(a-c)=(k-l)n

Similarly,
(b-d)=(p-q)n ... p,q ∈ ℤ⁺

Therefore
(a-c)≡(b-d) mod n