School of Mathematics and Statistics

MATH2400 FINITE MATHEMATICS

Trimester 2, 2021                                        TEST 2 (SAMPLE)                                        VERSION 2

● 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) Solve the simultaneous equations

with the additional condition that x is divisible by 3.

ii) Find all solutions with x in the interval [−193, 157].

2. [10 marks]

i) Use the Euler theorem to compute efficiently 7³²⁸⁶⁴⁷ (mod 55).

ii) Explain why we cannot use the same approach to compute 15²⁴³²¹⁶ (mod 55)?

iii) What is the largest integer n ≤ 57 such that has a primitive root.

3. [10 marks]

i) Let M = 36×57. Find the prime number factorisation of M and calculate the Euler function (M).

ii) The message (a, b, c, d) is encoded as c = (x, y, a, z, b, c, d) where x = a + b + d, y = a + c + d, z = b + c + d using the Hamming matrix:

a) Encode the message (1 1 0 1).

b) Assuming at most one error, correct and decode the received message r = (1 0 1 1 0 0 1).

END OF TEST