Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_
MATH1061 Assignment
All assignments in this course must be submitted electronically and SUBMITTED AS A SINGLE PDF FILE.
Prepare your assignment solutions using Word, LaTeX, Windows Journal, or other application, ensuring that
your name, student number and tutorial group number appear clearly at the top to the first page, and then
save your file in pdf format. Alternatively, you may handwrite your solutions and scan or photograph your
handwritten work to create a pdf file. Make sure that your pdf file is legible and that the file size is not
excessive. Use the assignment submission link in Blackboard to submit the pdf file.
1. (16 marks) Prove or disprove each of the following statements.
(a) For all d ∈ Z+, there exists n ∈ Z such that bn+1
d
c = dn
d
e.
(b) ∀n ∈ Z, if (3n + 2)2 6≡ 4 (mod 6) then n is odd.
(c) ∀n ∈ Z, if (2n + 1) ≡ 1 (mod 3) then n is odd.
(d) For all m,n, d ∈ Z+, if m | n and d - n then d - m.
(e) For all real numbers r and s, if rs
3
is irrational then r is irrational or s is irrational.
2. (2 marks) Use the Euclidean Algorithm to compute the greatest common divisor of −8765 and
1234. Show all of your steps.
3. (2 marks) Let n and m be positive integers, where the unique prime factorisations of n and m
are given by:
n = 2 · 3a · 72 · 13b for some fixed a, b ∈ Z+
m = 3k · 7 · 11 · 13` for some fixed k, ` ∈ Z+
If a < k and b > `, determine the greatest common divisor of n and m and the least common
multiple of n and m.
4. (5 marks) Consider the sequence {bn}n≥1 defined recursively as
b1 = 1, b2 = 3, b3 = 7 and bk = 2bk−1 + bk−3 for each integer k ≥ 4.
Use strong mathematical induction to prove that bn ≤ 3n−1 for each integer n ≥ 1.
Assignment continues next page.
MATH1061 (S1, 2023) Assignment 2 Due: 4pm on 3 April 2023
5. (5 marks) Consider the following proposal of a direct proof of a statement, given in the image
below. Do you agree with the arguments made in the proof? Is anything missing, or are there
any corrections to be made? Explain any potential issues with the text provided and then either
provide a new proof of your own or clearly explain why that is impossible.