Homework 2 - The Integers
Due Date: February 15, 2023
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Instructions
- Submit your file by 11:59 pm on the due date. Late submissions will not be accepted.
- If you'd like, you can use the following LaTeX Homework Template.
- In all assignments, give justifications for your answers. For example, if you claim that a certain relation is not an equivalence relation because it is not transitive then show that it is not transitive.
- Problems taken from Chapter 2 - The Integers
Problems
- Problem 6
- Problem 15: Only parts (a), (b), (c)
- Problem 16
- Problem 22
- Problem 27
- BONUS: Problem 19. I found a "solution" online to this problem. However, I think that it is incomplete and, if I were you, I would try another approach. Hint: The number 2 is the only even prime. Thus, if \(xy=a^2\) and \(a = p_1p_2\cdots p_n\) for primes \(p_i\) and with \(p_1\leq p_2\leq\cdots\leq p_n\), then either \(p_1=2\) or \(p_1 > 2\). Treat each case separately. The first case \(p_1=2\) is easy. For the second case use induction on the number \(n\) of primes. The second case just means that all the primes \(p_i\) are odd and every odd integer \(p\) is of the form \(p = 2q+1\). There's probably a shorter proof.