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 div d r=a mod d Solution 3 mod 7 Since 66 is larger than 3, we should be able to obtain 66 by consecutively adding 7 to 3 if 66 ≡ 3 mod 7. 3 mod 7 ≡ 3+7 mod 7 ≡ 10 mod 7 ≡ 10+7 mod 7 ≡ 17 mod 7 ≡ 17+7 mod 7 ≡ 24 mod 7 ≡ 24+7 mod 7 ≡ 31 mod 7 ≡ 31+7 mod 7 ≡ 38 mod 7 ≡ 38+7 mod 7 ≡ 45 mod 7 ≡ 45+7 mod 7 ≡ 52 mod 7 ≡ 52+7 mod 7 ≡ 59 mod 7 ≡ 59+7 mod 7 ≡ 66 mod 7 We then obtained that 66 is congruent to 3 mod 7