Theory question 114
Modular Arithmetic(Questions 14 and 15 are for candidates in Ghana, Sierra Leone and The Gambia only.)
(a) Copy and complete the multiplication table modulo 5 on the set {1, 2, 3, 4}.
(b) From your table; (i) solve the expression 2n * 4 ≡ 3; (ii) find the value of n for which 2 * (3 * n) ≡ 2.
(c) The functions f and g are defined as f : x → x − 2 and g : x → 2x2 − 1. Solve: (i) f(x) = g(−1/2); (ii) f(x) + g(x) = 0.
Show the model answer
(a) Rows (× 1, 2, 3, 4): 1: 1, 2, 3, 4; 2: 2, 4, 1, 3; 3: 3, 1, 4, 2; 4: 4, 3, 2, 1.
(b)(i) 2n * 4 = 8n ≡ 3n ≡ 3 (mod 5), so n = 1.
(ii) 2 * (3 * n) = 6n ≡ n ≡ 2, so n = 2.
(c)(i) g(−½) = 2(¼) − 1 = −½. x − 2 = −½ → x = 3/2.
(ii) x − 2 + 2x2 − 1 = 0 → 2x2 + x − 3 = 0 → (2x + 3)(x − 1) = 0 → x = 1 or x = −3/2.
In modulo 5, replace each product by its remainder on division by 5.