Problem 1
c.f. Problem 1.7
Suppose \(a,b\) are integers. Prove that, if \(a|b\) and \(b|a\), then \(a=b\) or \(a=-b\).
Assignment 1
Due Friday, September 2, 2022
Ordinarily, paper assignments are due at the beginning of class. Since class is still meeting virtually, you must submit your completed assignment by emailing the instructor by Friday, September 2 at 5:00 PM. If your assignment is handwritten, then you may submit a scanned document or a series of images taken with your phone. A PDF is preferred, but other formats are acceptable if the instructor can read them. If you anticipate any technical difficulties submitting the assignment, then email the instructor as soon as possible so alternate arrangements can be made.
Problem 2
c.f. Problem 1.12
Suppose \(a,b,c,d\) are integers. Prove that, if \(a|b\) and \(c|d\), then \(ac|bd\).
Problem 3
c.f. Problem 1.15(a)
Suppose \(a,b\) are integers. Prove that, if \(x^2+ax+b=0\) has an integer root, then the root divides \(b\).
Problem 4
c.f. Problem A.1
Prove that \[1^2 + 2^2 + \cdots + n^2 = \frac{n(2n+1)(n+1)}{6}\] for all positive integers.
Problem 5
c.f. Problem A.7
Suppose that \(a_0 = a_1 = 1\) and \(a_{n+1}=a_n + 2a_{n-1}\) for all positive integers \(n\). Prove that \[a_n = \frac{2^{n+1} + (-1)^n}{3}\] for all non-negative integers \(n\).
Problem 6
c.f. Problem A.14
Prove by induction that \(n^5-n\) is divisible by \(5\) for all positive integers \(n\).