Question

Prove that if m, d, and k are integers and d > 0, then (m + dk) mod d = m mod d.

Answer 1

From the rules of modular arithmetic, (A+B)mod C= (A)modC +(B)modC

So,  (m + dk) mod d =  (m) mod d+ (dk) mod d

clearly (dk) mod d = 0, as one of the things (dk) mod d represents, is the remainder of dk, when divided by d, which is clearly 0.


Thus

(m + dk) mod d =  (m) mod d+ (dk) mod d= (m) mod d+ 0=(m) mod d


(m + dk) mod d =(m) mod d