MATH1061 Assignment 2 Due 1pm Thursday 9 September 2021 This Assignment is compulsory, and contributes 5% towards your final grade. It should be submitted by 1pm on Thursday 9 September, 2021. In the absence of a medical certificate or other valid documented excuse, assignments submitted after the due date will receive late penalties as described in Section 5.3 of the Electronic Course Profile. Prepare your assignment as a pdf file, either by typing it, writing on a tablet or by scanning/photographing your handwritten work. Ensure that your name, student number and tutorial group number appear on the first page of your submission. Check that your pdf file is legible and that the file size is not excessive. Files that are poorly scanned and/or illegible may not be marked. Upload your submission using the assignment submission link in Blackboard. Please note that our online systems struggle with filenames that contain foreign characters (e.g. Chinese, Japanese, Arabic) so please ensure that your filename does not contain such characters. 1. (12 marks) Prove or disprove each of the following statements. (a) For any integer n, if n2 + 7 is not divisible by 4, then n is even. (b) For every n ∈ Z+, b3n3+1 n2 c = 3n. (c) ∀a, b ∈ Zodd, if a 6≡ b (mod 4) then ab−1 2 ∈ Zodd. (Note: Zodd represents the set of all odd integers.) 2. (3 marks) (a) Use the Euclidean Algorithm to determine the greatest common divisor of −7521 and 1635. (b) Determine the least common multiple of −7521 and 1635. 3. (3 marks) (a) Determine the unique prime factorisation of 59895 in standard form. Show your working. (b) Find the smallest positive integer n such that 59895n = a4 for some integer a. Explain your reasoning. 4. (4 marks) Generalise the proof given in Lecture 9 that √ 2 is irrational, in order to prove the following lemma. Lemma: For every prime p, √ p is irrational. In your proof, you should make use of the fact that for every integer n, and for every prime p, if p | n2 then p | n. (This a fact that can be proved using the prime factorisation theorem, but you do not need to prove this fact.) 5. (6 marks) Use mathematical induction to prove that 3 | (n3 + 5n + 6) for every integer n ≥ 0.