To find the modular inverse of \( x = 189^{-1} \mod 4 \), we need to find a number \( y \) such that:

\[
189 \times y \equiv 1 \pmod{4}
\]

### Step 1: Simplify \( 189 \mod 4 \)
First, compute \( 189 \mod 4 \):

\[
189 \div 4 = 47 \quad \text{(remainder = 1)}
\]

So:

\[
189 \equiv 1 \pmod{4}
\]

### Step 2: Set Up the Equation
We now have:

\[
1 \times y \equiv 1 \pmod{4}
\]

This simplifies to:

\[
y \equiv 1 \pmod{4}
\]

### Step 3: Identify the Inverse
The modular inverse \( y \) of \( 1 \) modulo \( 4 \) is simply:

\[
y = 1
\]

Thus:

\[
189^{-1} \equiv 1 \pmod{4}
\]

Would you like further details or have any more questions? Here are some related questions to deepen your understanding:

1. How can you find the modular inverse of any number modulo \( n \)?
2. What is the significance of the modular inverse in solving equations?
3. How would the process differ if the modulus was a prime number?
4. What happens if a number does not have a modular inverse?
5. How can the Extended Euclidean Algorithm be used to find modular inverses?
6. Why is the modular inverse unique modulo \( n \)?
7. How does finding a modular inverse relate to solving linear congruences?
8. Can modular inverses be used in cryptographic algorithms like RSA?

**Tip:** When finding modular inverses, simplifying the problem by reducing the number modulo \( n \) can often make calculations much easier.

Learn how to find the modular inverse of 189 modulo 4 using step-by-step calculations and understand the significance of modular inverses in mathematics. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation