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 (3x4mod19)(3x \equiv 4 \mod 19), we proceed as follows:

Step 1: Simplify the congruence

We need to solve for xx such that: 3x4(mod19).3x \equiv 4 \pmod{19}.

Step 2: Find the modular inverse of 3mod193 \mod 19

The modular inverse of 3mod193 \mod 19 is a number bb such that: 3b1(mod19).3b \equiv 1 \pmod{19}. We can find bb using the extended Euclidean algorithm. Performing the calculations, we find: b=13.b = 13.

Thus, 3113mod193^{-1} \equiv 13 \mod 19.

Step 3: Solve for xx

Multiply both sides of 3x4(mod19)3x \equiv 4 \pmod{19} by the modular inverse of 33: 133x134(mod19).13 \cdot 3x \equiv 13 \cdot 4 \pmod{19}. Simplify: x52(mod19).x \equiv 52 \pmod{19}. Reduce 52mod1952 \mod 19: x14(mod19).x \equiv 14 \pmod{19}.

Step 4: General solution

The solution to this modular equation is: x14(mod19).x \equiv 14 \pmod{19}. This means that all solutions are of the form: x=14+19k,kZ.x = 14 + 19k, \quad k \in \mathbb{Z}.

Do these solutions form an infinite set?

Yes, the solutions form an infinite set because kk can be any integer (positive, negative, or zero), and thus there are infinitely many integers xx satisfying the congruence.


Let me know if you need further details or explanations! Here are five related questions for deeper understanding:

  1. How is the modular inverse calculated using the extended Euclidean algorithm?
  2. What does it mean for a solution to a modular equation to be "unique modulo nn"?
  3. Can modular equations always be solved, or are there conditions for solvability?
  4. How do we handle modular equations with coefficients that are not coprime to the modulus?
  5. 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