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
