To solve the equation $1081 \cdot x + 3571 = 0$ in $\mathbb{Z}_{5395}$, we will proceed with modular arithmetic. Let me work out the solution.

The equation can be rewritten in modular form: $1081 \cdot x \equiv -3571 \pmod{5395}$

Next, we need the multiplicative inverse of $1081$ modulo $5395$ so that we can solve for $x$. Let's compute this.

sympy import mod_inverse

Define the constants

a = 1081 # Coefficient of x b = -3571 # Constant term (modular RHS) modulus = 5395 # Modulus

Compute the modular inverse of 1081 modulo 5395

inverse_a = mod_inverse(a, modulus)

Solve for x in the modular equation

x_solution = (inverse_a * b) % modulus inverse_a, x_solutionThe multiplicative inverse of $1081$ modulo $5395$ is $4856$. Using this, the solution to the modular equation is: $x \equiv 4149 \pmod{5395}$

Thus, the solution is $x = 4149$.

Would you like a step-by-step breakdown of how the modular inverse was calculated? Here are some related questions to deepen your understanding:

1. How is the modular inverse computed using the Extended Euclidean Algorithm?
2. What are the properties of modular arithmetic that make solving such equations possible?
3. Why is it necessary to use modular reduction in this equation?
4. How does modular arithmetic apply in cryptographic algorithms like RSA?
5. What happens if the coefficient $1081$ does not have a modular inverse?

Tip: Always ensure that the modulus and the coefficient are coprime when solving modular equations, as this guarantees the existence of a modular inverse.