Math Problem Statement

solve step by step: 4115 * x = 2 mod 18107

also cross verify

Solution

We are tasked with solving the following modular equation:

4115x2(mod18107)4115 \cdot x \equiv 2 \pmod{18107}

Step-by-Step Solution:

Step 1: Understand the Problem

We need to find xx such that:

4115x2(mod18107)4115 \cdot x \equiv 2 \pmod{18107}

This is a linear congruence, which can be solved by finding the multiplicative inverse of 41154115 modulo 1810718107, then multiplying both sides of the congruence by this inverse.

Step 2: Find the Multiplicative Inverse of 4115 Modulo 18107

To find the multiplicative inverse of 41154115 modulo 1810718107, we can use the Extended Euclidean Algorithm. The goal is to find an integer yy such that:

4115y1(mod18107)4115 \cdot y \equiv 1 \pmod{18107}

We will apply the Euclidean algorithm to 41154115 and 1810718107.

Euclidean Algorithm Steps:

  1. Divide 1810718107 by 41154115 and find the remainder.
  2. Repeat the process for the new pair (divisor and remainder) until the remainder is 1.

Let me calculate the inverse using the Euclidean algorithm: Implementing the Extended Euclidean Algorithm to find the inverse of 4115 mod 18107

def extended_gcd(a, b): # Base case: when b is 0, return a and coefficients for x and y if b == 0: return a, 1, 0 else: gcd, x1, y1 = extended_gcd(b, a % b) x = y1 y = x1 - (a // b) * y1 return gcd, x, y

Given values

a = 4115 b = 18107

Calculate gcd, x (inverse), and y using extended GCD

gcd, x_inverse, y = extended_gcd(a, b)

x_inverse % b # To ensure the inverse is positiveThe multiplicative inverse of 41154115 modulo 1810718107 is 36173617. This means:

411536171(mod18107)4115 \cdot 3617 \equiv 1 \pmod{18107}

Step 3: Multiply Both Sides of the Original Equation by 3617

Now, to solve for xx, we multiply both sides of the congruence 4115x2(mod18107)4115 \cdot x \equiv 2 \pmod{18107} by 36173617:

36174115x36172(mod18107)3617 \cdot 4115 \cdot x \equiv 3617 \cdot 2 \pmod{18107}

Since 361741151(mod18107)3617 \cdot 4115 \equiv 1 \pmod{18107}, this simplifies to:

x36172(mod18107)x \equiv 3617 \cdot 2 \pmod{18107}

Step 4: Compute the Value of xx

Now calculate 36172mod181073617 \cdot 2 \mod 18107:

x=36172=7234x = 3617 \cdot 2 = 7234

So, the solution to the congruence is:

x7234(mod18107)x \equiv 7234 \pmod{18107}

Step 5: Cross-Verify the Solution

Finally, let's cross-verify that x=7234x = 7234 satisfies the original equation 4115x2(mod18107)4115 \cdot x \equiv 2 \pmod{18107}.

41157234mod181074115 \cdot 7234 \mod 18107 Cross verify the solution x = 7234

Check if 4115 * x is congruent to 2 modulo 18107

result = (4115 * x) % 18107 result # Should be 2 if the solution is correctThe cross-verification shows that:

411572342(mod18107)4115 \cdot 7234 \equiv 2 \pmod{18107}

Thus, the solution x=7234x = 7234 is correct.

Final Answer:

The solution to the congruence 4115x2(mod18107)4115 \cdot x \equiv 2 \pmod{18107} is:

x7234(mod18107)x \equiv 7234 \pmod{18107}


Would you like further details on any step? Here are some related questions:

  1. How does the Extended Euclidean Algorithm work in general?
  2. What is the significance of the multiplicative inverse in modular arithmetic?
  3. How do you handle modular equations with larger numbers?
  4. What happens if there is no solution to a modular equation?
  5. Can the Euclidean algorithm be used for finding greatest common divisors of large numbers efficiently?

Tip: Always check if the numbers involved in a modular equation are coprime (gcd = 1) before trying to find a multiplicative inverse, as an inverse exists only when they are coprime!

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Math Problem Analysis

Mathematical Concepts

Modular Arithmetic
Linear Congruences
Euclidean Algorithm

Formulas

ax ≡ b (mod m)
Extended Euclidean Algorithm

Theorems

Multiplicative Inverse in Modular Arithmetic

Suitable Grade Level

Grades 10-12