10.8. Verifying the Solution

Try these first

1. 1.

   What is the solution for the linear congruence 4x ≡ 2 (mod 6)?

   Hint

   Divide both sides by 2 after checking GCD.

2. 2.

   If 5x ≡ 10 (mod 15), what are possible values of x?

   Hint

   What happens when you divide both sides by 5?

3. 3.

   What does it mean for a and N to be coprime when solving linear congruences?

   • They share a common divisor.
   • Their GCD is 1.
   • They are both even.

   Hint

   Recall the definition of coprime.

4. 4.

   True or False: Every linear congruence has a unique solution.

   • True
   • False

   Hint

   Consider the structure of modular arithmetic.

5. 5.

   Given five different congruences x ≡ 1 (mod 5), x ≡ 2 (mod 6), x ≡ 3 (mod 7), x ≡ 4 (mod 8), and x ≡ 5 (mod 9), determine the unique solution modulo the product of these moduli.

   Hint

   Start solving each congruence stepwise using pairwise relationships.

6. 6.

   Find the multiplicative inverse of 15 modulo 26. Show each step.

   Hint

   Remember the steps to apply the algorithm carefully.