Definitions Division algorithm Let a be an integer and d a positive integer. Then there are unique 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 Theorem 5 Let m be a positive integer. If a ≡ b(mod m) and c ≡ d(mod m), then a+c ≡ b+d(mod m) and ac ≡ bd(mod m). Solution a=11(mod 19) b ≡ 3(mod 19) 0 <= c <= 18 Use theorem 5: c ≡ a^(3)+4b^(3)(mod 19) =11^(3)+4*3^(3)(mod 19) =1331+4*27(mod 19) =1331+108(mod 19) =1439(mod 19) =14(mod 19) We then obtain c=14 with 0 <= c <= 18.