Practice Finding Solutions In A Specific Range (10.9) - Linear Congruence Equations and Chinese Remainder Theorem

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9 questions

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3 questions · about 2 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
What is the solution to 2x ≡ 4 (mod 3)?
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Find the integer value of `x` satisfying the equation.
Question 2.
Determine if 1x ≡ 5 (mod 10) has solutions, and if so, provide one.
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Solve the equation normally then check for modulo.
3Question 3.
What kind of equations are expressed as ax ≡ b (mod N)?
- Linear equations
- Linear Congruences
- Quadratic equations
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Look for terms that involve modulus.
4Question 4.
True or False: Every linear congruence has at least one solution.
- True
- False
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Consider the structure of congruences.
Question 5.
Prove that if ax ≡ b (mod N) has a solution for GCD(a, N) = d, then it implies a system of equations.
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Use properties of divisors and modular arithmetic.
Question 6.
For three congruences x ≡ 1 (mod 6), x ≡ 2 (mod 15), and x ≡ 3 (mod 10), find all solutions up to M.
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Work through each congruence step to consolidate solutions.

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