# Modular inverses

## What is an inverse?

Recall that a number multiplied by its inverse equals 1. From basic arithmetic we know that:

- The inverse of a number A is 1/A since A * 1/A = 1
**e.g. the inverse of 5 is 1/5** - All real numbers other than 0 have an inverse
- Multiplying a number by the inverse of A is equivalent to dividing by A
**e.g. 10/5 is the same as 10* 1/5**

**What is a modular inverse?**

In modular arithmetic we do not have a division operation. However, we do have modular inverses.

- The modular inverse of A (mod C) is A^-1
- (A * A^-1) ≡ 1 (mod C) or equivalently (A * A^-1) mod C = 1
- Only the numbers coprime to C (numbers that share no prime factors with C) have a modular inverse (mod C)

**How to find a modular inverse**

A

__naive method__of finding a modular inverse for A (mod C) is:**step 1.**Calculate A * B mod C for B values 0 through C-1

**step 2.**The modular inverse of A mod C is the B value that makes A * B mod C = 1

Note that the term B mod C can only have an integer value 0 through C-1, so testing larger values for B is redundant.

**Example: A=3 C=7**

**Step 1. Calculate **A * B mod C** for **B values __0 through C-1__****

3 * 0 ≡ 0 (mod 7)

3 * 1 ≡ 3 (mod 7)

3 * 2 ≡ 6 (mod 7)

3 * 3 ≡ 9 ≡ 2 (mod 7)

3 * 4 ≡ 12 ≡ 5 (mod 7)

3 * 5 ≡ 15 (mod 7) ≡

__1__(mod 7) <------ FOUND INVERSE!3 * 6 ≡ 18 (mod 7) ≡ 4 (mod 7)

**Step 2. The modular inverse of A mod C is the B value that makes **__A * B mod C = 1__

__A * B mod C = 1__

5 is the modular inverse of 3 mod 7 since 5*3 mod 7 = 1

Simple!

Let's do one more example where we don't find an inverse.

**Example: A=2 C=6**

**Step 1. Calculate A * B mod C for B values 0 through C-1**

2 * 0 ≡ 0 (mod 6)

2 * 1 ≡ 2 (mod 6)

2 * 2 ≡ 4 (mod 6)

2 * 3 ≡ 6 ≡ 0 (mod 6)

2 * 4 ≡ 8 ≡ 2 (mod 6)

2 * 5 ≡ 10 ≡ 4 (mod 6)

**Step 2. The modular inverse of A mod C is the B value that makes A * B mod C = 1**

No value of B makes A * B mod C = 1. Therefore, A has no modular inverse (mod 6).

This is because 2 is not coprime to 6 (they share the prime factor 2).

This is because 2 is not coprime to 6 (they share the prime factor 2).

**This method seems slow...**

There is a much faster method for finding the inverse of A (mod C) that we will discuss in the next articles on the Extended Euclidean Algorithm. First, let's do some exercises!