Thank you for a wonderful year and congratulations for sticking it out till the end! Here is your final task - a cooperative mission, see if you can beat it!

I misprinted the date on this assignment in the actual text but, just ignore that

Hello! This is the problem sheet for the Chinese Remainder Theorem :)

Our first week of number theory! The introductory concepts may be ones you have heard of before - they concern divisibility and factors, including things like prime numbers and gcd (greatest common divisor). That said, we will derive all of our information from these definitions plus algebra basics (manipulating equations) and logic (that we developed last semester), so you need to know how to work with these definitions and follow the proofs of the theorems. You should spend some time on this sheet, as these fundamentals will be important.

As we ended our geometry unit today, I have assigned some problems to get you back in the habit of working with formal logical language. I definitely want you to attempt at least the first three logic problems; take a look at the rest of the problems on this sheet as well, if they interest you. Don't be turned off by long paragraphs - formal logic requires reading comprehension, so I expect you to read what I write for you in these problems.

I've changed the style of the homeworks - problems are now all approximately equal in difficulty, and I want each of you to prepare to present one of them the next week

As there are less of us in class now, I decided we can try something a little more dynamic, so here I introduce you to the Elephants and Hamsters theory. This is a fun little study in axiomatic theory, and includes some rather nontrivial, sometimes difficult, combinations of the axioms in order to solve the problems. Look through the sheet, and attempt problems under the "Practice Problems" section. Solving these problems will require you to keep careful track of what you are talking about, and to be able to distinguish the different categories (Elephants, Hamsters, theme songs) so that you be able to understand how to simultaneously work with objects that have different properties.

These are mostly all problems from past homework sheets, so not much you haven't seen before. The exception is problem 10, which is not a problem we proved or discussed in class. Work through any of these problems that you didn't get a chance to do in past weeks, and try number 10 if you want a challenge.

Please attempt any five of the problems on this homework! :)

Hello! Please work through problems 1-5 at least, and then choose whatever you want from the remaining problems

Please do problems 6-12.

Please do problems 1-6; of the remaining problems 8-10 and Miscellaneous 1-3, please choose one of the six to do. Additionally, please consider registering for the AMC Competition if you are interested! The registration link is in an announcement, viewable from the schoolnova website homepage.

Please do all the problems before the New York State lottery problem, except for the problem about five-letter words; beyond that, please attempt one more combinatorics problem, and one of the miscellaneous problems. As always, if you wish to do more, please do, I will look at everything you submit.