11.2.3. Uniqueness Proof Part for the Chinese Remainder Theorem

Try these first

1. 1.

   Explain Euclid's Lemma in your own words.

   Hint

   Think of how primes work with products.

2. 2.

   What does it mean for numbers to be 'pairwise relatively prime'?

   Hint

   Look at pairs of numbers and their factors.

3. 3.

   What does Euclid's Lemma state?

   • A prime divides a product of integers if it divides each integer separately.
   • A prime divides a product of integers if it divides at least one of the integers.
   • A prime only divides even integers.

   Hint

   Think of how primes operate in multiplication.

4. 4.

   True or False: If a system of linear congruences has more than one solution, they must be congruent modulo M.

   • True
   • False

   Hint

   Consider the implications of congruence properties.

5. 5.

   Given three integers 4, 6, and 10, demonstrate using CRT that a linear combination of these can yield a unique solution within a defined modulus that is their product.

   Hint

   Break down each modulus and product relation carefully.

6. 6.

   Suppose there are solutions x1 and x2 for a given set of linear equations. Show that if these solutions yield differing results, it contradicts the uniqueness principle established in CRT.

   Hint

   Bring in the notion of divisibility and modulo relations.