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Inverse Modulo Calculator

Inverse Modulo Calculator

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Inverse Type:

a * x = 1 mod m

Integer (a):

Modulo (m):

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Use this inverse modulo calculator to calculate the modular inverse of an integer. So let’s move on and discuss this tricky concept in detail and check how this free calculator will help us to speed up our calculations.

Stay focused!

What Is Inverse Modulo?

In the light of applied mathematics:

“A particular integer number x is said to be ad the inverse modulo of a random integer a if it yields the identity element after performing certain mathematical operations from x to a”

To understand the tricky concept of the inverse modulo, you must be aware of the modulo congruence explained in the upcoming section. So just stay focused!

Modulo Congruence:

Whenever in mathematical calculations the word Congruent is seen, this means there is some equivalency being described in the phenomenon.

Likewise, we have the following scenario to describe the congruence in case of inverse modulo:

  • Suppose you have three mathematical numbers.
  • Among these, two numbers are integers named as x and y and one is named natural number n

Now the integers x and y will be considered congruent to each other if:

  • Both yield the same remainder when divided by the natural number n

Also, we have another approach in this case:

If the difference of the integers x and y (x-y) yields zero when divided by the natural number n, they are said to be equivalent of each other

Generic Representation:

a ≡ b (mod n)

Our free inverse modulo calculator with steps also displays the final answer in the generic form mentioned above.

Types of Inverse Modulo:

Depending upon the operation being used on the integers x and a, there are a couple of inverse modulo types described as under:

Additive Inverse Modulo:

We all are familiar with the additive identity which is 0. Now when it comes to additive inverse modulo that could also be determined by using this inverse modulo calculator in seconds, we have the following situation: (Keep in mind that the condition must be fulfilled)

a + x ≡ 0 (mod m)

Let’s elaborate the above expression to understand it better:

  • In the above expression, the integer number x is considered the additive inverse modulo of a if a + x and 0 both become equivalent to the modulo given.

Multiplicative Inverse Modulo:

Just like additive identity, the multiplicative identity is 1. Coming to the point, the modular multiplicative inverse of any number satisfies the expression as defined below:

a * x ≡ 1 mod m

The above expression elaborates that:

  • The integer number x is considered the multiplicative inverse modulo of a if a * x and 1 both become equivalent to the modulo given.
Methods to Determine the Inverse Multiplicative Modulo:

As far as the analysis of multiplicative modular inverse is concerned, we have various approaches to determine it. These include:

The Naive Method:

This is indeed the simplest method to determine the modular multiplicative inverse of a number. But on the other hand, it includes a lengthy analysis. Let’s discuss further!

{0, …, m – 1}

In this method, we are required to perform division of a * x by modulo number m. The combination that yields the remainder 1 is considered the multiplicative modular inverse combination. No doubt this is a very lengthy process and that is why we advise you using our free inverse modulo calculator with steps.

Extended Euclidean Algorithm:

This particular method take into consideration the Bezout’s Identity that states:

“Make a supposition that you are having four integers divided into two groups as:

  • a and b
  • x and y

Now these sets of integers are able to follow the Bezout’s identity if:

a * x + b * y = gcd(a, b)”

So let’s move on and learn how to determine the modular multiplicative inverse using this identity:

What you need to keep in mind here is that:

  • Integer a and modulo m must be coprime and their cumulative greatest common factor must be 1, such that:

gcd(a,m) = 1

Following the Bezout’s identity, we have:

a * x + m * y = 1

Now moving towards analysis of the multiplicative modular inverse on the basis of the data above mentioned:

We need to apply the mod operation on both sides of the equation [mod(m)]

This allows us to know that:

m * y ≡ 0 (mod m)

The above expression means that o modulo m must be congruent to the multiples of integers. And it helps us to determine the following equation:

a * x ≡ 1 (mod m)

Fermat’s Little Theorem:

This operation considers a couple of facts mentioned below:

  • a is not the multiple of the integer m
  • Also, m is not a prime number of a

Statement:

Keeping in view the above mentioned theory, Fermat introduced new way of calculating multiplicative modular inverse of numbers which is as under:

“If a is not factored by m in case m is prime, then you can easily divide \(a^{m-1} – 1\) by th integer m”

By taking into consideration the above statement, we can write:

\(a^{m-1} ≡ 1 mod m\)

Here the interesting fact to know is that whatever the method you choose for calculations, our best modular inverse calculator with steps will satisfy the results calculated from each method.

How Inverse Modulo Calculator Works?

Let this free modulo inverse calculator determine the modular inverse within a few clicks. Let’s have a look at the steps that you must follow to operate this calculator. Stay with it!

Input:

  • From the first drop-down list, select whether you want to calculate the “Multiplicative Inverse”or “Additive Inverse”
  • After you make a selection, go for entering the value of the integer and modulus in their respective fields
  • At last, tap the calculate button

Output:

The free inverse solver does the following calculations:

  • Calculates the additive inverse modulo
  • Also calculates the multiplicative inverse modulo
  • Displays the step by step calculations

FAQ’s:

What is the inverse of 7 modulo 26?

The inverse of the given combination of integers is 15 that you can also verify by subjecting to the best inverse modulo calculator.

What do you mean by the additive inverse?

As the name justifies that we need to add a negative number of the same number that we are required to determine the additive inverse of.

Generic Representation:

a +(-a) = 0

What is the inverse of 13 modulo 2436?

The inverse modulo of the given set of integers is 927. For instance, you can also verify the results by putting the values in our free online mod inverse calculator.

Conclusion:

The concept of inverse modulo is worth considering as it aids in determining the solutions to the linear system of congruences. And this is why we have developed this inverse modulo calculator with steps to calculate the exact inverse in terms of modulus for any integer number.

References:

From the source of Wikipedia: Modular multiplicative inverse, Modular arithmetic, Integers modulo m, Computation, Euler’s theorem, Applications

From the source of Khan Academy: Modular inverses, Fast modular exponentiation, Modular multiplication, Modular exponentiation, The Euclidean Algorithm

From the source of Lumen Learning: Characteristics of Inverse Functions, Domain and Range