Homework 1

1. Divisibility tests
   • State and prove 2 divisibility tests different from the ones discussed in class. Provide multiple proofs whenever you can.
   • Which of your proofs can be taught in middle school? In high school?
   • Which proofs require more mathematics than (usually) is studied in high school? Could these be discussed in a math club? What (additional) definitions/theorems would you have to provide then?

   Suggestions: some good/relatively easy numbers to choose from are 4, 5, 6, 8, 10, 11, 12, 16, 25, 100, but there are others as well.

   Resources: Many divisibility tests are posted on Internet. E.g. type "divisibility tests" in Google. However, most pages do not provide proofs. Try to prove the rules you recall or find on your own. If you get stuck on some, I encourage you to consult your number theory textbook, work together, and/or ask me.

2. Bases other than 10
   • Convert 2015₆ to base 10.
   • Convert 2015 (base 10) to base 6.
   • Convert 12012₃ to base 4.
   • If a number is represented as 1234 in base a and as 1020304 in base b, what can you say about bases a and b?

3. Look through some of the Math Field Day problems (High school Mad Hatter and Leap Frog) and previous problem solving sessions topics. Suggest 2 topics for this year's sessions. Choose topics that (1) are represented at the Math Field Day, (2) ones where can teach concepts or problem solving strategies beyond the standard high school material or do problems of above the standart high school difficulty, and (3) you would like to teach. Make sure that the topics are not so narrow that we won't be able to find enough Math Field Day problems on them but not too broad either (so that we can concentrate on some concepts and problem solving strategies).