10.4. Statement of the Chinese Remainder Theorem

Try these first

1. 1.

   What is the general solution form of a linear congruence?

   Hint

   Consider how many solutions exist once you find one.

2. 2.

   Define pairwise coprime.

   Hint

   Think about how many numbers share divisors.

3. 3.

   What is the form of solutions for linear congruences?

   • A + kN
   • A - kN
   • k + N

   Hint

   Think about the structure of solutions.

4. 4.

   True or False: The Chinese Remainder Theorem guarantees a unique solution if the moduli share a common factor.

   • True
   • False

   Hint

   Recall the conditions for application of CRT.

5. 5.

   Demonstrate the application of the Chinese Remainder Theorem by forming and solving a system of congruences for a fictional age scenario.

   Hint

   Set up equations reflecting given ages and moduli.

6. 6.

   Given the congruences x ≡ 1 (mod 6), x ≡ 5 (mod 9), and x ≡ 3 (mod 10), apply the CRT to find the smallest positive solution.

   Hint

   Use the method of constructing equations based on defined remainders.