Math Problem Statement
Find all solutions of the equation (
3 4 x
) mod 1 9 . Do these solutions form an infinite set and why?
Solution
To solve the modular equation , we proceed as follows:
Step 1: Simplify the congruence
We need to solve for such that:
Step 2: Find the modular inverse of
The modular inverse of is a number such that: We can find using the extended Euclidean algorithm. Performing the calculations, we find:
Thus, .
Step 3: Solve for
Multiply both sides of by the modular inverse of : Simplify: Reduce :
Step 4: General solution
The solution to this modular equation is: This means that all solutions are of the form:
Do these solutions form an infinite set?
Yes, the solutions form an infinite set because can be any integer (positive, negative, or zero), and thus there are infinitely many integers satisfying the congruence.
Let me know if you need further details or explanations! Here are five related questions for deeper understanding:
- How is the modular inverse calculated using the extended Euclidean algorithm?
- What does it mean for a solution to a modular equation to be "unique modulo "?
- Can modular equations always be solved, or are there conditions for solvability?
- How do we handle modular equations with coefficients that are not coprime to the modulus?
- What are some applications of solving modular equations in number theory or cryptography?
Tip: Modular inverses exist only when the coefficient and the modulus are coprime. Always check their greatest common divisor before proceeding!
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Math Problem Analysis
Mathematical Concepts
Modular Arithmetic
Congruences
Number Theory
Formulas
ax ≡ b (mod n)
Extended Euclidean Algorithm for Modular Inverse
Theorems
Bezout's Theorem
Modular Inverse Theorem
Suitable Grade Level
Grades 10-12