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What is an inverse? 

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

  • The inverse of 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(mod 7)

3 * 4 ≡ 12(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


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 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!