Practice Discrete Mathematics (11.1) - Uniqueness Proof of the CRT - Discrete Mathematics

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6 sample questions

Question 1.
Define a linear congruence. Give an example.
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Remember the format of the equation.
Question 2.
What does it mean for two integers to be co-prime?
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Think about their divisibility.
3Question 3.
What does the Chinese Remainder Theorem guarantee?
- No solutions
- Unique solution modulo the product of moduli
- Multiple solutions within bounds
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Think about how CRT resolves multiple equations.
4Question 4.
True or False: If two integers are co-prime, they have a common divisor greater than 1.
- True
- False
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Consider the definition;
Question 5.
Prove that if a system of n linear congruences has a solution, there is a unique solution modulo the product of the moduli.
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Incorporate proof techniques like induction and explore implications of congruences.
Question 6.
Given the numbers 10, 12, and 15 with a known sum, find numbers congruent to each, ensuring the CRT principles are satisfied.
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Break down the system and consider pairwise co-primality.

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