MATH7861数学求解

Semester 1, 2022 MATH7861: Assignment 2
Assignment 2 must be submitted by 1 pm on Monday the 11th of April, 2022.
Submit your assignment using the TurnItIn link on the MATH1061/7861 Blackboard site.
For MATH7861 students this assignment is worth 10% of your final mark. Questions 1 to 6 will
be marked out of 30 and the essay question will be marked out of 15. Giving a total of 45. Note
that when the mark is entered into the blackboard site it will be scaled back to a mark out 30.

1. (6 marks) Answer each of the following questions, and in each case fully justify your
   answers.
   (a) If x and y are integers, is 9 a factor of 3x · 15y?
   (b) Suppose x is an integer such that
2. · 3 · 4 · 5 · x = 59 · 58 · 57 · 56 · 55.
   Write down the prime factorization of the product 59 · 58 · 57 · 56 · 55 and use it to
   justify your answers to the following questions.
   (i) Does 59 | x?
   (ii) Does 29 | x?
   (iii) Does 118 | x?
3. (4 marks) Let a, b, c be any integers. For each of the following statements, if it is true
   prove it or if it is false provide a counterexample.
   (i) If d | a and d | b, then gcd(a, b) = d.
   (ii) If a | b and b | c, then c | a.
   (iii) If b ≡ 0(mod a) and c ≡ 0(mod b), then c ≡ 0(mod a).
4. (4 marks) Prove the following statement.
   For any integer n ≥ 2, n
5. − 3 is never divisible by 4. (Hint consider n odd and then n
   even.)
6. (4 marks) Use the Euclidean algorithm to calculate gcd(1494, 672).
7. (6 marks) Prove that, for all n ≥ 1,
   Xn
   i=1
   (5i − 4) = n(5n − 3)
   2
   .
8. (6 marks) Use Mathematical Induction to prove that, for all integers n ≥ 5,
   4n + 1 < 2
   n
   .
9. (15 marks)
   Short Essay ON
   – Venn Diagrams
   – Hasse Diagrams
   You are asked to choose from the above topics and write a 1.5 to 2.5 page essay that
   discusses the related theory and applications of the topic you have chosen. If you wish
   you can compare and contrast the two topics or just discuss one. Your discussion should
   include historical aspects of the topic and how this topic relates to the mathematics studied
   in MATH7861. Examples and diagrams of the relevant mathematics should be included.
   Some references to get you started on the topic are available through the blackboard site.
   Essay Structure
   The essay should be TYPED with a minimum of 1.5 pages and a maximum of 2.5 pages
   including a bibliography. The font should be 12pt and margins should be around 1.5 cm.
   Overall the length of your essay should be around 1500 words, with diagrams to support
   your text. You can (but not required to) follow the following structure for you essay:
   Background+History (1-2 paragraphs): In this section you can introduce the main
   subject of this essay, discussing the historical aspects of the subject. (4 marks)
   Analysis Section (3-4 paragraphs): What are the mathematical ideas associated with
   this topic and how do they relate to the mathematics being studied in MATH7861?
   Can you give examples of how this mathematics is relevant in our ever day lives? (5
   marks)
   Conclusion (1 paragraph): Conclude the essay by summarizing the ideas presented, if
   possible discuss the impact of this mathematics on your current studies. (3 marks)
   Bibliography: At least 3 sources (you can include the supplied article as a reference)
   correctly cited in the essay and using a consistent citation format.