10.2. Solving Linear Congruences using Extended Euclid's Algorithm

Try these first

1. 1.

   What are the conditions under which a linear congruence has a unique solution?

   Hint

   What do we mean by the coefficients being coprime?

2. 2.

   Solve the linear congruence 4x ≡ 2 (mod 6).

   Hint

   Check if a and n share common factors.

3. 3.

   What does it mean for two numbers to be coprime?

   • They have no common divisors
   • They are equal
   • They are both prime

   Hint

   Think about the GCD definition.

4. 4.

   True or False: The Chinese Remainder Theorem can find solutions for any moduli.

   • True
   • False

   Hint

   Consider the requirements for CRT to work.

5. 5.

   Solve the system of congruences: x ≡ 1 (mod 2), x ≡ 2 (mod 3), x ≡ 3 (mod 4). Show your work and verify all solutions.

   Hint

   Break down each congruence systematically.

6. 6.

   Apply the Extended Euclidean Algorithm to determine the inverse of 14 modulo 33, then solve 14x ≡ 11 (mod 33).

   Hint

   Look for steps in the algorithm outlining the GCD.