School of Mathematics and Statistics

MATH2400 FINITE MATHEMATICS

Trimester 2, 2021                                        TEST 000                                        VERSION 3

● Time Allowed: 30 minutes + 15 minutes for scanning and uploading

● Total Number of Marks: 30

● For each question you must show your working; unsubstantiated answers will not gain positive marks

● Submit each question separately, clearly marking the files Q1, Q2, Q3

1. [10 marks]

i) Without solving the following system of simultaneous congruences

estimate how many integer solutions x ∈ [0, 10⁵ ] it has. An error up to 10% is acceptable.

ii) Can we apply the Chinese Remainder Theorem directly to solve the following system of congruences

You must justify your answer.

2. [10 marks]

i) Using that 2¹⁷ ≡ 11 (mod 21), show that 21 is composite. You must justify your answer.

ii) Let p be an odd prime. Find another integer n with (n) = (p).

3. [10 marks]

i) What is the number of primitive roots modulo 29?

ii) Consider a code with encodes 0 and 1 as 000 and 010, respectively. How many errors can this code correct? Give a full justification of your conclusion.

END OF TEST