Homework for Week 3

Part 1

  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 2

  1. Find all integers x that solve the following congruences:
    1. 4x+7 = 5 mod 9
    2. 4x+5 = 1 mod 10
    3. 6x+5 = 8 mod 10
    4. 12x = 3 mod 17
    5. 10x = 4 mod 18
  2. What is a zero-divisor? What are the zero-divisors mod 27? What are the zero-divisors mod 24?
  3. What is a unit? What are the units mod 27? What are the units mod 24?
  4. What is the (multiplicative) inverse of 8 mod 27? What is the (multiplicative) inverse of 7 mod 24?
  5. 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)