Homework 1

This assignment is due by the beginning of class on Wednesday, September 3^{rd}. 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 $k$. (Note: I made a small change to the problem since the original posting. The old version can also be solved, but this is a little more precise, so it's a better result to carry around in your toolbox.)
- From Strayer, do the following problems:
- 5
- 10(a)
- 11(b),(d)
- 18
- 21(b),(d)
- 23
- 29

It would also be good to get some practice on other problems, even though I won't be collecting them. For this, I'd suggest you take a look at the following problems from Strayer: 1(c), 1(d), 2, 3(b), 12, 25, 30. Any other problems you want to work on are also good practice, so if these problems all seem to easy, try out some others on your own.

page revision: 14, last edited: 29 Oct 2008 19:02

Heyy! Having some trouble with proving the first part. I broke it down into three different cases and proved that it held true for all the cases. My cases were as follows:

1) n > k

2) n < k

3) n = k

Anyone else go about this problem differently??

Also, on the homework, I thought it was very helpful on problem 11 to prove that any even plus any odd is equal to an odd number. After proving that, I used that claim to prove the other parts.

Well anyway, if you could provide any input on the first problem, that would be very helpful. Thanks!

~Alex

ReplyOptionsTry using the division algorithm on n, with d=k. Then you just need to figure out how to show that for each possible r, you can find an element in the set that would be divisible by k.

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