Theory question 108
Modular Arithmetic(a) Copy and complete the following table for multiplication modulo 11.
⊗: 1, 5, 9, 10
1: 1, 5, 9, 10
5: 5, __, __, __
9: 9, __, __, __
10: 10, __, __, __
Use the table to: (i) evaluate (9 ⊗ 5) ⊗ (10 ⊗ 10); (ii) find the truth set of I. 10 ⊗ m = 2; II. n ⊗ n = 4.
(b) When a fraction is reduced to its lowest term, it is equal to ¾. The numerator of the fraction when doubled would be 34 greater than the denominator. Find the fraction.
Show the model answer
(a) Row 5: 5, 3, 1, 6. Row 9: 9, 1, 4, 2. Row 10: 10, 6, 2, 1.
(i) 9 ⊗ 5 = 1 and 10 ⊗ 10 = 1, so the answer is 1 ⊗ 1 = 1.
(ii) I. 10 ⊗ m = 2 → m = 9, truth set {9}. II. n ⊗ n = 4 → n = 9, truth set {9}.
(b) Let the fraction be 3k/4k. 2(3k) = 4k + 34 → k = 17. The fraction is 51/68.
Each entry is the remainder when the ordinary product is divided by 11.