Homework 1

This assignment is due by the beginning of class on Friday, January 30^{th}. Be sure to review the homework guidelines before getting started.

- Prove that for any $n \in \mathbb{Z}$ and any positive integer
*k*, at least one number from the set $\{n,n+1,\cdots, n+k-1\}$ is divisible by*k*. - Prove that if
*a,b*and*c*are integers satisfying $a \mid b$ and $a \nmid c$, then $a \nmid b+c$. - From Strayer, do the following problems:
- 2
- 3(b,d)
- 5
- 11(b),(d)
- 13
- 32(f)
- 33(d)
- 36(a)
- 54(f)

If you want some extra practice on problems from these sections, here are a few suggestions. You do *not* need to turn these in.

- Suppose that
*a*is a nonzero integer and*d*is an integer so that $d \mid a$. Prove that $|d| \leq |a|$. - From Strayer, Chapter 1: 3(a,c,e,f), 10(a,c), 12, 42, 54(a-e), 55