Posted by **Samantha** on Saturday, December 4, 2010 at 3:21pm.

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

- Discrete Math -
**MathMate**, Saturday, December 4, 2010 at 7:45pm
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

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