(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.
(Rotman 1.95) Find all solutions to the linear system x = 12 mod 25
x = 2 mod 30.
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.
Which of the following are valid bit-parity-check codewords (explain): 0110101, 010110, 1101101101?
Which of the following are valid ISBN codes (explain): 0-14-235213-9, 11-2463-253-7, 0-471-43809-X?
Write the permutation (134)(25) in one-line notation, arrow notation, and two-line notation.
Write the permutation 21435 in two-line notation, cycle notation, and arrow notation.
Let u = (134)(25) and v = 21435. Compute uv and vu. Compute the inverse of u and the inverse of v.