$20x \equiv 24 \mod 164$ and $x \equiv 15 \mod 73$ solve the given system for x

To solve the system of congruences:

1. $ 20x \equiv 24 \mod 164 $
2. $ x \equiv 15 \mod 73 $

### Step 1: Solve the first congruence $ 20x \equiv 24 \mod 164 $
We start by simplifying this congruence. Divide both sides by the greatest common divisor (GCD) of 20 and 164, which is 4:

\[
\frac{20x}{4} \equiv \frac{24}{4} \mod \frac{164}{4}
\]

This simplifies to:

\[
5x \equiv 6 \mod 41
\]

Next, we need to solve this linear congruence. We find the multiplicative inverse of 5 modulo 41. The inverse $ y $ satisfies:

\[
5y \equiv 1 \mod 41
\]

We can find $ y $ using the extended Euclidean algorithm, which gives:

\[
y = 33 \quad \text{since} \quad 5 \times 33 = 165 \equiv 1 \mod 41
\]

Now, multiply both sides of \( 5x

Learn how to solve the system of modular congruences 20x ≡ 24 (mod 164) and x ≡ 15 (mod 73) step-by-step. This problem involves advanced modular arithmetic and the extended Euclidean algorithm to find solutions. Suitable for advanced high school and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation