11.2.6. Application of Chinese Remainder Theorem

Try these first

1. 1.

   What is the Chinese Remainder Theorem?

   Hint

   Consider its relation to moduli.

2. 2.

   Define coprime numbers.

   Hint

   Think of prime numbers.

3. 3.

   What is guaranteed by the Chinese Remainder Theorem?

   • No solution exists
   • A unique solution exists
   • Multiple solutions exist

   Hint

   Think about the uniqueness in systems of equations.

4. 4.

   True or False: If two numbers are congruent mod m and m is prime, they are equal.

   • True
   • False

   Hint

   Consider the possibilities outside of congruence.

5. 5.

   Prove that the CRT can be extended to more than two moduli.

   Hint

   Think about the relationships of congruences.

6. 6.

   If 17 and 19 are coprime moduli, find a number less than 323 that is congruent to 5 mod 17 and congruent to 12 mod 19.

   Hint

   Calculate the respective products and their inverses.