a) Find all x Z 143 such that x²7 (mod 143)

.

b) Is 15²⁰¹⁹1 divisible by 9?

c) Find xZ ₁₀₁ such that 8^x2 (mod 101).. In your solution you will likely need to use the FLT and will have to compute the inverse of 3 modulo 100. Do you see an equally simple way to find xZ₁₀₁ such that 8^x5 (mod 101)? If not, why not? (Hint: This is related to the discrete logarithm problem.)

d)

Let us assume that known plaintext attack on an LFSR cipher revealed the following part of the key:

1 1 1 0 0 1 0 1 1 1. . .

Find the probable recurrence relation generating that key.

e) Find all xZ₂₂₁ such that

x2 (mod 13)

x15 (mod 17)

.