## Introduction of Mod Inverse Calculator:

Mod inverse calculator is a digital tool that is used to **find the inverse modulo** of a given gcd (a, b) number to find the value of integer x. It computes both the additive or multiplicative inverse modulo of given values in less than a minute.

Although inverse modulo is a complex method to solve the integer and helps to get the inverse modulo for multiplication or addition. Our inverse modulo calculator provides the step by step solution of inverse modulo for the arithmetic operation.

## What is Inverse Modulo?

Inverse modulo is a **set theory method** that is used to find the multiplicative inverse or additive inverse of a given modulo. It is defined as an integer which is a coprime number when it uses the arithmetic operation to get a remainder number 1 (for multiplication) or 0 (for additive) using the given mod number.

In mathematics, it is represented with a congruent sign (≡) which means equivalent. On the left side of the congruent sign is the coprime integers. On the right side mod value m and the inverse number for addition 0 or multiplication is 1.

$$ a \;≡\; b \;(mod\; n) $$

### For Multiplicative Inverse Modulo:

$$ a \times n \;≡\; 1 \;mod\; m $$

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

### For Multiplicative Inverse Modulo:

$$ a + n \;≡\; 0 \;mod\; m $$

$$ gcd\;(a,m) \;=\; 0 $$

To perform these calculations efficiently, you can use the mod inverse calculator for the modular arithmetic inverse.

## Working of Inverse Modulo Calculator:

The modular inverse calculator uses an easy method to solve the complicated problems of modular arithmetic inverse for both addition and multiplication.

If you calculate modulo inverse by yourself it will take so much time to **get the inverse number x** but you do not need to worry as in this modulo inverse calculator you give the input number n and mod value.

When you give the input, the inverse mod calculator analyzes the given data and searches which number gives the best result of inverse modulo. Let's use the below example to understand the how the calculator calculate modular inverse in some steps.

**Step 1**:

Identify the given data, as you can see in the below example a = 101 and modulo = 4620.

**Step 2**:

For addition use this rule (**a + n ≡ 0 mod m**), or for multiplication use this rule ( a × n ≡ 1 mod m).

**Step 3**:

For additive inverse modulo, we find the number “n” that is added with the “a” number and give the inverse value zero.

**Step 4**:

In the given example x = 4519 so add 101 give 4620 number after addition. This number gives the remainder value zero when you take the difference of a + n - mod(m) = x^{-1}. That means 4519 is the inverse modulo of mod(4620).

**Step 5**:

For multiplicative inverse, we choose a number n when you multiply with a it gives b as a × n = b. When b is divided by mod(m) then the remainder must be 1.

As you see in the below example mod inverse calculator takes x = 1601 and multiplies with 101 as 1601 × 101 = 161701. When it **divides 161701 by 4620** it gives them remainder 1. It means 1601 is the inverse modulo of mod(4620).

## Does 101 have an Inverse Modulo 4620?

Yes, 101 has an inverse modulo 4620 for both **addition or multiplication inverse values**. For the calculation both the inverse solution is given below,

### For Additive Inverse Modulo:

To find the additive inverse modulo we use this formula as,

$$ a + n \;≡\; 0\; mod\; m $$

After putting the given data the above equation becomes,

$$ 101 +\; x \;=\; 0\; mod \;4620 $$

$$ Put\; x\; = \; 4519 $$

After putting the x value we get 101 + 4519 = 4620

So the above equation becomes,

$$ 101 + 4519 \;=\; mod\; 4620 $$

$$ 4620 \;=\; 0\; mod\; 4620 $$

$$ gcd\; (101,4519) \;=\; 4620 $$

So inverse modulo of 4620 is 4620.

### For Multiplicative Inverse Modulo

To find the multiplicative inverse modulo we use this formula as,

$$ a \times n\; ≡ \;1 mod\; m $$

After putting the given data the above equation becomes,

$$ 101 \times x \;=\; 1 \;mod\; 4620 $$

$$ Put\; x \;=\; 1601 $$

After putting the x value we get 101 × 1601 = 161701. So the above equation becomes,

$$ 101 \times 1601 \;=\; 1 \;mod\; 4620 $$

$$ 161701 \;=\; 1\; mod\; 4620 $$

$$ gcd \; (4620,101) \;=\; 1601 $$

So inverse modulo of 4620 is 1601.

## How to Find the Additive Inverse of 3 Modulo 7?

To find the additive inverse modulo we use this formula as,

$$ a + n \;≡\; 0\; mod\; m $$

After putting the given data the above equation becomes,

$$ 3 + x \;=\; 0 \;mod\; 7 $$

$$ Put\; x \;=\; 4 $$

After putting the x value we get 3 + 4 = 7. So the above equation becomes

$$ 3 + 4 \;=\; 0\; mod \; 7 $$

$$ 7\; =\; 0\; mod\; 7 $$

$$ gcd\; (7,3) \;=\; 4 $$

So the inverse modulo of 7 is 4.

## How to Find Inverse 17 Modulo 19?

### For Addition Inverse Modulo:

To find the additive inverse modulo we use this formula as,

$$ a + n \;≡\; 0\; mod\; m $$

After putting the given data the above equation becomes,

$$ 17 + x \;=\; 0\; mod\; 19 $$

$$ Put\; x\; =\; 2 $$

After putting the x value we get 17 + 2 = 19. So the above equation becomes,

$$ 17 + 2 \;=\; mod\; 19 $$

$$ 19 \;=\; 0\; mod\; 19 $$

$$ gcd \;(19,17) \;=\; 2 $$

### For Multiplicative Inverse Modulo:

To find the multiplicative inverse modulo we use this formula as,

$$ a \times n \;≡\; 1\; mod\; m $$

After putting the given data the above equation becomes,

$$ 17 \times x \;=\; 1\; mod\; 19 $$

$$ Put\; x \;=\; 9 $$

After putting the x value we get 17 × 9 = 153. So, the above equation becomes

$$ 17 \times 9 \;=\; 1\; mod\; 19 $$

$$ 153 \;=\; 1\; mod\; 19 $$

$$ gcd\; (19,17) \;=\; 9 $$

So inverse modulo of 19 is 9.

## How to Find Mod Inverse Calculator?

The inverse modulo calculator has a simple layout that enables you to solve inverse modulo problems immediately. You do not need to put any external effort just follow our instructions so that you do not find any difficulty during the calculations. These steps are:

**Enter the value**of “a” in the input field.- Enter the value of mod m in the input field.
- Check your input number before clicking the calculate button to get the actual solution.
- Click the “Calculate” button for the solution of an inverse modulo questions.

Press the “Recalculate” button for more evaluation of the inverse modulo example solution. - You can use the load example to do more evaluation process of the inverse modulo questions with a solution.

## Outcome of Modular Inverse Calculator:

The mod inverse calculator gives you a **solution of your input number** when you click on the calculate button that starts the process. It may include as:

**Result Box**:

You get the solution to find the inverse modulo problems when you click on the result button.

**Steps Box**

Click on the Possible steps option to get the solution of the given inverse modulo question from this calculator in a step-by-step method

## Advantages of Using Modulo Inverse Calculator:

The inverse mod calculator is an amazing tool for solving modulo problems using inverse modulo method. It gives you serval benefits whenever you use it for finding the unknown number x. These benefits are:

- The inverse modulo calculator is a reliable tool as it gives you
**accurate solutions**to inverse modulo problems - It has a user-friendly tool that anyone or even a beginner can easily use to solve the inverse to find x number
- The modular inverse calculator does not ask to sign up before using it for the calculation.
- It is a speedy tool that provides you with the solution to the given number for an inverse modulo question in a fraction of a second
- The calculator is a free online tool, you can find it for evaluation of inverse modulo questions without spending any fee.
- The mod inverse calculator can be used for practice to solve inverse modulo examples of different numbers with a solution.