Summarize this article:
105 LearnersLast updated on September 11, 2025
Inverse Modulo Calculator
Calculators are reliable tools for solving simple mathematical problems and advanced calculations like trigonometry. Whether you’re cooking, tracking BMI, or planning a construction project, calculators will make your life easy. In this topic, we are going to talk about inverse modulo calculators.
What is an Inverse Modulo Calculator?
An inverse modulo calculator is a tool to find the modular multiplicative inverse of a given integer under a specified modulus.
This mathematical concept is crucial in number theory and cryptography. The calculator simplifies the process of finding the inverse, saving time and effort.
How to Use the Inverse Modulo Calculator?
Given below is a step-by-step process on how to use the calculator:
Step 1: Enter the integer and modulus: Input the number and the modulus into the respective fields.
Step 2: Click on calculate: Click on the calculate button to find the inverse and get the result.
Step 3: View the result: The calculator will display the result instantly.
How to Calculate the Inverse Modulo?
To calculate the inverse modulo, a simple formula is used. If 'a' is the integer and 'm' is the modulus, the inverse is a number 'x' such that: a * x ≡ 1 (mod m)
This means that 'a' and 'm' must be coprime for the inverse to exist. The Extended Euclidean Algorithm is often used to find the inverse.
Tips and Tricks for Using the Inverse Modulo Calculator
When using an inverse modulo calculator, there are a few tips and tricks to make it easier and avoid mistakes: Ensure that the integer and modulus are coprime, as the inverse does not exist otherwise.
Be familiar with the properties of modular arithmetic. Use the calculator for large numbers to avoid manual errors.
Common Mistakes and How to Avoid Them When Using the Inverse Modulo Calculator
Despite using a calculator, mistakes can happen. Here are some common ones to be aware of:
Mistake 1
Not checking if the integer and modulus are coprime.
Always verify that the greatest common divisor (GCD) of the integer and modulus is 1.
If not, the inverse does not exist.
Mistake 2
Inputting incorrect values.
Ensure that the values entered are correct, as small mistakes can lead to incorrect results.
Mistake 3
Misinterpreting the result.
Remember that the result is valid under the given modulus.
It must satisfy the condition a * x ≡ 1 (mod m).
Mistake 4
Relying on the calculator too much without understanding the theory.
Understanding the basics of modular arithmetic and the Euclidean Algorithm can help in interpreting results correctly.
Mistake 5
Assuming the inverse exists for any integer and modulus.
The inverse only exists if the integer and modulus are coprime.
Check this condition before using the calculator.
Inverse Modulo Calculator Examples
Problem 1
What is the inverse of 3 modulo 11?
Use the Euclidean Algorithm: 3 * x ≡ 1 (mod 11) The inverse of 3 modulo 11 is 4, since 3 * 4 = 12 and 12 ≡ 1 (mod 11).
Explanation
Using the Euclidean Algorithm, we find that the inverse of 3 under modulus 11 is 4, as it satisfies the equation 3 * 4 ≡ 1 (mod 11).
Problem 2
Find the inverse of 7 modulo 26.
Use the Euclidean Algorithm: 7 * x ≡ 1 (mod 26) The inverse of 7 modulo 26 is 15, since 7 * 15 = 105 and 105 ≡ 1 (mod 26).
Explanation
The calculation shows that multiplying 7 by 15 gives a result that satisfies the condition for an inverse under mod 26.
Problem 3
Determine the inverse of 10 modulo 17.
Use the Euclidean Algorithm: 10 * x ≡ 1 (mod 17) The inverse of 10 modulo 17 is 12, since 10 * 12 = 120 and 120 ≡ 1 (mod 17).
Explanation
The inverse is found to be 12, as it satisfies the equation 10 * 12 ≡ 1 (mod 17).
Problem 4
What is the inverse of 5 modulo 23?
Use the Euclidean Algorithm: 5 * x ≡ 1 (mod 23) The inverse of 5 modulo 23 is 14, since 5 * 14 = 70 and 70 ≡ 1 (mod 23).
Explanation
The result shows that 5 multiplied by 14 meets the required condition under mod 23.
Problem 5
Find the inverse of 4 modulo 9.
Use the Euclidean Algorithm: 4 * x ≡ 1 (mod 9) The inverse of 4 modulo 9 is 7, since 4 * 7 = 28 and 28 ≡ 1 (mod 9).
Explanation
The inverse is found to be 7, as it satisfies the condition for an inverse under mod 9.
FAQs on Using the Inverse Modulo Calculator
1.How do you calculate the inverse modulo?
2.Does an inverse always exist?
3.Why are coprime integers necessary for the inverse?
4.How do I use an inverse modulo calculator?
5.Is the inverse modulo calculator accurate?
Glossary of Terms for the Inverse Modulo Calculator
- Inverse Modulo: A value that, when multiplied with the original integer, results in 1 under a specific modulus.
- Coprime: Two numbers are coprime if their greatest common divisor (GCD) is 1.
- Euclidean Algorithm: A method for finding the greatest common divisor of two numbers.
- Modular Arithmetic: A system of arithmetic for integers, where numbers "wrap around" after reaching a certain value, the modulus.
- Extended Euclidean Algorithm: An extension of the Euclidean Algorithm that finds the coefficients of Bézout's identity.
Explore More calculators
![Important Math Links Icon]()
Previous to Inverse Modulo Calculator
![Important Math Links Icon]()
Next to Inverse Modulo Calculator
Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
