Practice Finding A Special Linear Combination For The Solution (10.6) - Linear Congruence Equations and Chinese Remainder Theorem

Practice: Finding a Special Linear Combination for the Solution

Lesson 10.6Linear Congruence Equations and Chinese Remainder Theorem

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10 questions

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6 sample questions

Question 1.
Solve the linear congruence 4x ≡ 2 (mod 10).
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Try dividing both sides if GCD of coefficients is 1.
Question 2.
What is the multiplicative inverse of 3 mod 7?
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Find a number which, when multiplied by 3, gives a remainder of 1 when divided by 7.
3Question 3.
What does the term 'linear congruence' refer to?
- An algebraic equation
- A system of equations
- An equation under modular arithmetic
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Think about the context of modular operations.
4Question 4.
True or False: The Chinese Remainder Theorem guarantees a unique solution modulo the product of the pairwise co-prime moduli.
- True
- False
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Recall the main properties of CRT.
Question 5.
Prove that the solutions for the linear congruence 12x ≡ 12 (mod 18) include infinite entries.
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How can you express x based on multiples of the reduced expression?
Question 6.
Given three congruences x ≡ 4 (mod 7), x ≡ 5 (mod 9), and x ≡ 6 (mod 10), find the smallest x that satisfies them.
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Combine each congruence carefully, noting pairwise properties.

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