MATH1061 Assignment 2 Due 10am Monday 2 September 2019This Assignment is compulsory, and contributes 5% towards your final grade. It must be submitted by10am on Monday 2 September, 2019. In the absence of a medical certificate or other valid documentedexcuse, assignments submitted after the due date will not be marked.Submission You will receive a coversheet for this assignment by email. Print that coversheet, stapleit to the front of your assignment (which may be handwritten) and submit your assignment using theassignment submission system which is located in the corridor between buildings 69 and 62.1. (16 marks) Prove or disprove each of the following statements.(a) There exists a prime number x such that x + 16 and x + 32 are also prime numbers.(b) ∀a, b, c,m ∈ Z+, if a ≡ b (mod m), then ca ≡ cb (mod m).(c) For any positive odd integer n, 3|n or n2 ≡ 1 (mod 12).(d) There exist 100 consecutive composite integers.2. (5 marks) Prove the following statement using a proof by contradiction.For any prime number p, p > 10 implies 7|(p2 + 3)(p2 + 5)(p2 + 6).3. (5 marks) Prove the following statement using a proof by contraposition.∀r ∈ Q, s ∈ R, if s is irrational, then r + 1sis irrational.4. (3 marks) Use the Euclidean algorithm to determine the greatest common divisor of 2288 and4875.5. (6 marks) Use mathematical induction to prove that for each integer n ≥ 4, 5n ≥ 22n+1 + 100.
