10.6. Finding a Special Linear Combination for the Solution

Try these first

1. 1.

   Solve the linear congruence 4x ≡ 2 (mod 10).

   Hint

   Try dividing both sides if GCD of coefficients is 1.

2. 2.

   What is the multiplicative inverse of 3 mod 7?

   Hint

   Find a number which, when multiplied by 3, gives a remainder of 1 when divided by 7.

3. 3.

   What does the term 'linear congruence' refer to?

   • An algebraic equation
   • A system of equations
   • An equation under modular arithmetic

   Hint

   Think about the context of modular operations.

4. 4.

   True or False: The Chinese Remainder Theorem guarantees a unique solution modulo the product of the pairwise co-prime moduli.

   • True
   • False

   Hint

   Recall the main properties of CRT.

5. 5.

   Prove that the solutions for the linear congruence 12x ≡ 12 (mod 18) include infinite entries.

   Hint

   How can you express x based on multiples of the reduced expression?

6. 6.

   Given three congruences x ≡ 4 (mod 7), x ≡ 5 (mod 9), and x ≡ 6 (mod 10), find the smallest x that satisfies them.

   Hint

   Combine each congruence carefully, noting pairwise properties.