Homework for Week 3

Part 1

Problems 10-14 from Fundamental Theorem of Arithmetic Lecture.

Part 2

1. What is the remainder after dividing 10¹⁰⁰ by 7? (Rotman 1.81; 10¹⁰⁰ is a google)
2. Prove that if n is a number (written base 10) and n' is obtained from n by rearranging its digits, then n-n' is divisible by 9. (For instance, if n were 245643, then n' could be 445632.) (Rotman 1.79)
3. Compute all powers mod n, for n=2,3,4,5,6,7,8. In other words, find xⁱ for all x and all i, for n=2,3,4,5,6,7,8. Give at least two observations/conjectures based on your data.
4. Compute 3⁴² mod 15.
5. Compute 15²² mod 7.
6. Compute 2⁶⁸ mod 21.

• Part 1: Due Tuesday 2/7
• Part 2: Due Friday 2/10