Practice Verifying The Solution (10.8) - Linear Congruence Equations and Chinese Remainder Theorem

Practice: Verifying the Solution

Lesson 10.8Linear Congruence Equations and Chinese Remainder Theorem

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10 questions

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6 sample questions

Question 1.
What is the solution for the linear congruence 4x ≡ 2 (mod 6)?
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Divide both sides by 2 after checking GCD.
Question 2.
If 5x ≡ 10 (mod 15), what are possible values of x?
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What happens when you divide both sides by 5?
3Question 3.
What does it mean for a and N to be coprime when solving linear congruences?
- They share a common divisor.
- Their GCD is 1.
- They are both even.
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Recall the definition of coprime.
4Question 4.
True or False: Every linear congruence has a unique solution.
- True
- False
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Consider the structure of modular arithmetic.
Question 5.
Given five different congruences x ≡ 1 (mod 5), x ≡ 2 (mod 6), x ≡ 3 (mod 7), x ≡ 4 (mod 8), and x ≡ 5 (mod 9), determine the unique solution modulo the product of these moduli.
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Start solving each congruence stepwise using pairwise relationships.
Question 6.
Find the multiplicative inverse of 15 modulo 26. Show each step.
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Remember the steps to apply the algorithm carefully.

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