Math Problem Statement
Solution
The problem asks you to solve the linear congruence:
Step-by-Step Solution:
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Simplify the equation mod 69: Notice that , since . So the equation simplifies to:
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Find the multiplicative inverse of 5 mod 69: To solve , we need to find the inverse of 5 modulo 69. This can be done using the extended Euclidean algorithm.
Applying the algorithm: Working backward, we find: So, the inverse of 5 mod 69 is , since .
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Solve for x: Multiply both sides of the simplified congruence by the inverse of 5, which is 14: Calculate: Now reduce 98 modulo 69:
So, the solution is:
Final Answer:
Let me know if you need more details or explanations!
Here are 5 related questions to deepen your understanding:
- What is the general method to find the inverse of a number in modular arithmetic?
- How would the solution change if the modulus were different, say mod 70?
- Can all linear congruences be solved using the method outlined?
- What if the coefficient of and the modulus are not coprime? How do you handle that case?
- How does the extended Euclidean algorithm work, and why is it useful in modular arithmetic?
Tip: When solving linear congruences, always check if the greatest common divisor (GCD) of the coefficient of and the modulus is 1, as this ensures that an inverse exists.
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Math Problem Analysis
Mathematical Concepts
Linear Congruences
Modular Arithmetic
Extended Euclidean Algorithm
Formulas
74x ≡ 7 (mod 69)
Inverse of a number modulo n
Theorems
Modular Inverses
Extended Euclidean Algorithm
Suitable Grade Level
Grades 11-12