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 ≡ 2a^(2)+3b^(2)(mod 19) =2*11^(2)+3*3^(2)(mod 19) =2*121+3*9(mod 19) =242+27(mod 19) =269(mod 19) =3(mod 19) We then obtain c=3 with 0 <= c <= 18.