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:
Step-by-Step Solution:
Step 1: Understand the Problem
We need to find such that:
This is a linear congruence, which can be solved by finding the multiplicative inverse of modulo , 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 modulo , we can use the Extended Euclidean Algorithm. The goal is to find an integer such that:
We will apply the Euclidean algorithm to and .
Euclidean Algorithm Steps:
- Divide by and find the remainder.
- 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 modulo is . This means:
Step 3: Multiply Both Sides of the Original Equation by 3617
Now, to solve for , we multiply both sides of the congruence by :
Since , this simplifies to:
Step 4: Compute the Value of
Now calculate :
So, the solution to the congruence is:
Step 5: Cross-Verify the Solution
Finally, let's cross-verify that satisfies the original equation .
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:
Thus, the solution is correct.
Final Answer:
The solution to the congruence is:
Would you like further details on any step? Here are some related questions:
- How does the Extended Euclidean Algorithm work in general?
- What is the significance of the multiplicative inverse in modular arithmetic?
- How do you handle modular equations with larger numbers?
- What happens if there is no solution to a modular equation?
- 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
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