Practice Construction Of The Solution X (10.7) - Linear Congruence Equations and Chinese Remainder Theorem

Practice: Construction of the Solution x

Lesson 10.7Linear 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 x in 5x ≡ 15 (mod 10)?
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Check numbers that give a remainder of 5 when multiplied by 5.
Question 2.
If 3x ≡ 6 (mod 9), what are possible values for x?
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Look for x that, when multiplied by 3, gives a remainder of 6.
3Question 3.
What method do we use to solve ax ≡ b (mod N) confidently?
- A. Simple division
- B. Extended Euclid's Algorithm
- C. Graphical method
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Think about which method guarantees the necessary inverse.
4Question 4.
True or false: The Chinese Remainder Theorem guarantees a unique solution to every set of congruences.
- True
- False
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Consider the conditions of the theorem.
5Question 5.
Solve the system of congruences using CRT:
- x ≡ 1 (mod 5)
- x ≡ 4 (mod 7)
- x ≡ 6 (mod 11).
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Think through each modulus contributions, expressing them individually.
Question 6.
Prove that if ax ≡ b (mod N) can be solved, then gcd(a, N) divides b.
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Consider the implications of solutions to mod forms.

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