Homework 1

1. Divisibility tests
   • State and prove at least 4 divisibility tests different from the one 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 3, 4, 5, 6, 8, 9, 10, 11, 12, 16, 25, 100, but there are a few others as well. E.g. do you know a divisibility test for 7?

   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 2011₆ to base 10.
   • Convert 2011 (base 10) to base 6.
   • Convert 2011₅ to base 7.
   • 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?