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 37 is larger than 3, we should be able to obtain 37 by consecutively 7 to 3 if 37 ≡ 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 We then note that 3 mod 7 is ≡alent with 31 and 38. 3 mod 7 is then not ≡alent with 37, since 31<37<38. Note 37 mod 7 ≡ 2 mod 7