Homework for Week 3
Part 1
- What is the remainder after dividing 10100 by 7? (Rotman 1.81; 10100 is a google)
- 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)
- 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.
- Compute 342 mod 15.
- Compute 1522 mod 7.
- Compute 268 mod 21.
Part 2
- Find all integers x that solve the following congruences:
- 4x+7 = 5 mod 9
- 4x+5 = 1 mod 10
- 6x+5 = 8 mod 10
- 12x = 3 mod 17
- 10x = 4 mod 18
- What is a zero-divisor? What are the zero-divisors mod 27? What are the zero-divisors mod 24?
- What is a unit? What are the units mod 27? What are the units mod 24?
- What is the (multiplicative) inverse of 8 mod 27? What is the (multiplicative) inverse of 7 mod 24?
- Prove that if p is a prime and if a2=1 mod p then a = 1 mod p or a = -1 mod p. (Rotman 1.88)
- Part 1: Due Tuesday 2/9
- Part 2: Due Friday 2/12