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. Also, I encourage you to consult your number theory textbook, work together, and/or ask me.
Bases other than 10
Convert 20167 to base 10.
Convert 2016 (base 10) to base 7.
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?
State and provide a brief explanation for a few divisibility tests in base 12. (Try to find as many as you can.)