## Introduction to Multiplicative Modular Inverse Calculator

Multiplicative modular inverse calculator is an online tool that helps you to find the multiplicative inverse modulo in a few seconds. Our tool **finds the unknown number** x to get the solution of multiplicative inverse of a given number.

Our modular multiplicative inverse calculator is beneficial for anyone who want to find the gcd value of modulo having remainder 1. Hence, our tool is helpful for finding the multiplicative inverse modulo solution easily.

## What is a Modular Multiplicative Inverse

Multiplicative inverse modulo is defined as the number that gives the **remainder 1 value** when you divide a number with its modulo. It is noted that not every number has a multiplicative inverse.

Multiplicative inverse modulo exist in the numbers that have a relative prime number. The numbers having no relative prime number (can not give remainder 1) is not the multiplicative inverse modulo. Different types of set theory methods are used to solve the multiplicative inverse modulo problems.

## Principle of Multiplicative Inverse Modulo:

The **principles of multiplicative inverse modulo** on which multiplicative modular inverse calculator works and solve the tricky problems is,

$$ a \;≡\; b \;(mod\; n) $$

$$ a \times n \;≡\; 1\; mod\; m $$

$$ gcd\; (a,m) \;=\; 1 $$

Here,

- a and n: two coprime numbers
- mod m: the modulo
- gcd: the greatest common denominator

## How to Find Multiplicative Inverse Modulo?

Multiplicative inverse modulo problems can be determined with the **Euclid algorithm method**. It needs a coprime number n and v for finding the multiplicative inverse modulo problems.

Although it is a little tricky method to solve modulo problems, however the multiplicative inverse modulo calculator uses some simple steps to solve it.

**Step1:**

First, you check the given data from the question to find the multiplicative inverse modulo.

**Step2:**

Second, we have to find the n number which is multiplied by the given prime number v.

**Step3:**

Remember, you should choose n number that is the coprime with v.

**Step4:**

After finding the coprime multiplication, divide it by the modulo number.

**Step5:**

If it gives remainder 1 after division, then your n number is the multiplicative inverse of the given modulo.

Note: You do not always get a number that gives remainder 1 after division. Only the coprime number gives a multiplicative inverse if it is not the coprime number then you cannot find the modulus inverse.

## What is Multiplicative Inverse of 20 with Arithmetic Modulo 73?

For the calculation of the multiplicative inverse of 20 modulo 73, First, Multiplicative modular inverse calculator **find a number n** that is multiplied by the given number:

$$ 20 \times n \;≡\; 1\; (mod \; 73) $$

The above principle clearly shows when n is multiplied by 20, it gives a remainder of 1 while coprime is 73.

As per the Extended Euclidean Algorithm method, modulo multiplicative inverse calculator chooses different numbers using the trial and Error Method to find the prime number that gives the remainder 1.

Let's start with n number, n = a, a2, a3,....

**For n = 4**:

$$ 20 \times 4 \;≡\; (mod\; 73) $$

$$ 20 \times 4 \;≡\; -7\; (mod \; 73) $$

**For n = 8**:

$$ 20 \times 8 \;≡\; (mod\; 73) $$

$$ 20 \times 8 \;≡\; -14\; (mod\; 73) $$

**For n = 9**:

$$ 20 \times 9 \;≡\; (mod\; 73) $$

$$ 20 \times 9 \;≡\; -34\; (mod\; 73) $$

**For n = 8**:

$$ 20 \times 10 \;≡\; (mod\; 73) $$

$$ 20 \times 10 \;≡\; -54\; (mod\; 73) $$

**For n = 11**:

$$ 20 \times 11 \;≡\; (mod\; 73) $$

$$ 20 \times 11 \;≡\; 1\; (mod\; 73) $$

x = 11 is the multiplicative inverse of 20 modulo 73.

## What is the Multiplicative Inverse of 7 Modulo 31?

For the calculation of multiplicative inverse of 7 modulo 31, First, multiplicative inverse calculator modulo **find a number n** that is multiplied by the given number such that

$$ 7 \times n \;≡\; 1\; (mod\; 31) $$

The above principle clearly shows when n is multiply by 7, it gives a remainder of 1 while coprime is 31.

As per the Extended Euclidean Algorithm method, multiplicative modular inverse calculator chooses different numbers using the trial and Error Method to find the exact prime number that gives the remainder 1.

Let's start with n number,n = 1,2,3,.....n

**For n = 5**:

$$ 7 \times 5 \;≡\; (mod\; 31) $$

$$ 7 \times 5 \;≡\; 4\; (mod\; 31) $$

**For n = 6**:

$$ 7\times 6 \;≡\; (mod\; 31) $$

$$ 7 \times 6 \;≡\; 11\; (mod\; 31) $$

**For n = 7**:

$$ 7 \times 7 \;≡\; (mod\; 31) $$

$$ 7 \times 7 \;≡\; 18\; (mod\; 31) $$

**For n = 8**:

$$ 7 \times 8 \;≡\; (mod \;31) $$

$$ 7 \times 8 \;≡\; 25\; (mod\; 31) $$

**For n = 9**:

$$ 7 \times 9 \;≡\; (mod\; 31) $$

$$ 7 \times 9 \;≡\; 1\;(mod\; 31) $$

n = 9 is the multiplicative inverse of 7 modulo 31.

It is written as:

$$ 1 \;≡\; 7 \times 9\; (mod\; 31) $$

$$ 1 \;≡\; 63\; (mod\; 31) $$

## What is the Multiplicative Inverse of 4 Modulo 11?

For the calculation of the multiplicative inverse of 4 modulo 11, First, multiplicative modular inverse calculator finds a number n that is multiplied by the given number:

$$ 4 \times n \;≡\; 1\; (mod\; 11) $$

The above principle clearly shows when n is multiplied by 4, it gives a remainder of 1 while coprime is 11.

As per the **Extended Euclidean** Algorithm method, modular multiplicative inverse calculator chooses different numbers using the trial and Error Method to find the exact prime number that gives the remainder 1.

Let's start with n number,n = 1,2,3,.....n

**For n = 1**:

$$ 4 \times 1 \;≡\; (mod\; 11) $$

$$ 4 \times 1 \;≡\; -7\; (mod\; 11) $$

**For n = 2**:

$$ 4 \times 2 \;≡\; (mod\;11) $$

$$ 4 \times 2 \;≡\; -3(mod\;11) $$

**For n = 3**:

$$ 4 \times 3 \;≡\; (mod\; 11) $$

$$ 4 \times 3 \;≡\; 1 \;(mod\; 11) $$

n = 3 is the multiplicative inverse of 4 modulo 11.

It is written as,

$$ 1 \;≡\; 4 \times 3 \;(mod\; 11) $$

$$ 1 \;≡\; 12\; (mod\; 11) $$

## How to Use the Multiplicative Modular Inverse Calculator?

Multiplicative inverse modulo calculator contain a user friendly interface so that you can use it to calculate the multiples of numbers. You just need to follow some simple steps, which are:

**Enter the number**in the input box.- Enter the modulus value in the second input box.
- Review your input value before clicking the calculate button to start the calculation process.
- Click on the “Calculate” button to get the desired result of inverse multiplicative modulo.
- If you want to check our modulo multiplicative inverse calculator first then you can use the load example option.
- Recalculate button get a new page for solving more inverse modulo number values.

## Results from Modular Multiplicative Inverse Calculator:

Multiplicative modular inverse calculator gives you the **solution of inverse modulo number** problem when you give it an input. It gives you soluions in steps, which contain as:

**Result Option**:

You can click on the result option, it provides you with a solution of multiplicative inverse modulo problems.

**Possible Steps**:

When you click on the possible steps option it provides you step by step solution of inverse modular.

## Benefits of Multiplicative Inverse Modulo Calculator:

Multiplicative inverse calculator modulo gives you tons of benefits whenever you use it to calculate the inverse modulo problems. These advantages are:

- Our tool
**saves your time**and effort from doing complex calculations of the given number. - Modular multiplicative inverse calculator is a free-of-cost tool so you can use it to find the inverse modulo of numbers.
- It is a versatile tool that allows you to solve the inverse multiplicative number.
- You can use this modulo multiplicative inverse calculator for practicing multiplicative inverse modulo problems.
- It is a reliable tool that provides you accurate solutions whenever you use it to calculate a given inverse modulo problem.
- It provides a solution in steps so you get the solution in detail without missing any bit of its solution.
- Multiplicative modular inverse calculator is an easy-to-use tool so you do not need to make any external effort, just add the input number and get the solution.