Let d be a fixed positive integer greater than one If a b be

Let d be a fixed positive integer greater than one. If a, b belongs to Z, prove that a mod d = b mod d if and only if d | (a - b).

Solution

If a mod d = k, then a= dp+k , where p is an integer . Also, if b mod d , then b = dq +k’ where q is an integer. If a mod d = b mod d, then k = k’ so that a-dp = b-dq or, a-b = dp-dq = d(p-q). This implies that d|(a-b).

Let a mod d = s and b mod d = t. Then a = dp+s and b = dq+t where p, q are integers. Then a –b= dp+s- dq -t = d(p-q) +s-t .. However, since d|(a-b),hence s-t = 0 or s = t i.e a mod d = b mod d.