11.2. Uniqueness Proof of the CRT

Try these first

1. 1.

   What is Euclid's Lemma?

   Hint

   Think about primes and their properties.

2. 2.

   Explain divisibility in simple terms.

   Hint

   Consider how many times one number fits into another.

3. 3.

   What does Euclid's Lemma state?

   • A prime divides a sum.
   • A prime divides a product.
   • A prime divides an average.

   Hint

   Think about the definition of prime numbers.

4. 4.

   True or False: The Chinese Remainder Theorem guarantees unique solutions under all modulus conditions.

   • True
   • False

   Hint

   Consider what kind of moduli we discussed.

5. 5.

   Demonstrate through an example why Euclid's Lemma holds true by picking random integers and checking their products.

   Hint

   Use simple calculations to verify your findings.

6. 6.

   Find the unique solution of the following system: x ≡ 1 (mod 12), x ≡ 3 (mod 18).

   Hint

   Test values sequentially based on given equations.