Definitions Division algorithm Let a be an integer and d a positive integer. Then there are nique integers q and r with 0<= r < d such that a=dq+r q is called the quotient and r is called the remainder q=a/d r=a mod d Solution 3 mod 7 Since -67 is smaller than 3, we should be able to obtain -67 by consecutively subtracting 7 from 3 if -67≡ 3 mod 7. 3 mod 7 ≡ 3-7 mod 7 ≡ -4 mod 7 ≡ -4-7 mod 7 ≡ -11 mod 7 ≡ -11-7 mod 7 ≡ -18 mod 7 ≡ -18-7 mod 7 ≡ -25 mod 7 ≡ -25-7 mod 7 ≡ -32 mod 7 ≡ -32-7 mod 7 ≡ -39 mod 7 ≡ -39-7 mod 7 ≡ -46 mod 7 ≡ -46-7 mod 7 ≡ -53 mod 7 ≡ -53-7 mod 7 ≡ -60 mod 7 ≡ -60-7 mod 7 ≡ -67 mod 7 We then obtained that -67 is congruent to 3 mod 7