Practice Introduction To Linear Congruences (10.1) - Linear Congruence Equations and Chinese Remainder Theorem

Practice: Introduction to Linear Congruences

Lesson 10.1Linear Congruence Equations and Chinese Remainder Theorem

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10 questions

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6 questions · about 4 min · pass mark 70%. Hints available.
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4 questions · about 3 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
What is the form of a linear congruence?
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Look for the equation that involves a modulus.
Question 2.
Can a linear congruence have infinitely many solutions?
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Consider the adjustments involving k.
3Question 3.
What is a linear congruence?
- A polynomial equation
- An equation in modular arithmetic
- A geometric representation
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Focus on the definition involving moduli.
4Question 4.
True or False: The Chinese Remainder Theorem assures a unique solution for any system of linear congruences.
- True
- False
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Consider the conditions under which CRT applies.
Question 5.
Find all integer solutions for the linear congruence $8x \equiv 16 \mod{48}$ and explain your steps.
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Consider how to simplify the equation.
Question 6.
Prove that if $a$ is a solution of the system of congruences from the Chinese Remainder Theorem, then any number of the form $a + lm$ (where $m$ is the product of modulus) is also a solution.
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Write out what happens for each modulus.

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