Homework for Week 4

Part 1

  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)

Part 2

  1. (Rotman 1.91) Find all solutions to the simultaneous congruences:
    1. x=2 mod 5 and 3x = 1 mod 8;
    2. 3x=2 mod 5 and 2x=1 mod 3.
  2. (Rotman 1.95) Find all solutions to the linear system
    x = 12 mod 25
    x = 2 mod 30.
  3. Show that if a = r0 + 1000 r1+10002 r2 + 10003r3+ ... then a is divisible by 7 if and only if r0 - r1+r2-r3+... is divisible by 7.
  4. Which of the following are valid bit-parity-check codewords (explain): 0110101, 010110, 1101101101?
  5. Which of the following are valid ISBN codes (explain): 0-14-235213-9, 11-2463-253-7, 0-471-43809-X?
  6. Write the permutation (134)(25) in one-line notation, arrow notation, and two-line notation.
  7. Write the permutation 21435 in two-line notation, cycle notation, and arrow notation.
  8. Let u = (134)(25) and v = 21435. Compute uv and vu. Compute the inverse of u and the inverse of v.