Inverse Modulo Calculator
Expert Guide to Calculating Inverse Modulo
Module A: Introduction & Importance
Inverse modulo is a fundamental concept in modular arithmetic, crucial in number theory and cryptography. It helps us find a number that, when multiplied by another number, gives a result that is congruent to 1 modulo a third number.
Module B: How to Use This Calculator
- Enter the number (a) and modulus (m).
- Click ‘Calculate’.
- View the result and chart below.
Module C: Formula & Methodology
The inverse modulo of ‘a’ under modulus ‘m’ is the number ‘b’ such that (a * b) % m = 1. We use the extended Euclidean algorithm to find ‘b’.
Module D: Real-World Examples
Example 1
Find the inverse modulo of 3 under 11. (a = 3, m = 11)
Using the extended Euclidean algorithm, we find b = 4. So, (3 * 4) % 11 = 12 % 11 = 1.
Module E: Data & Statistics
| a | m | Inverse Modulo |
|---|---|---|
| 5 | 11 | 9 |
| 7 | 17 | 13 |
Module F: Expert Tips
- Inverse modulo is not unique. If one inverse exists, there are exactly ‘m – 1’ inverses.
- Inverse modulo is used in RSA encryption, a widely-used algorithm in secure communication.
Module G: Interactive FAQ
What if the inverse modulo doesn’t exist?
If the greatest common divisor (gcd) of ‘a’ and ‘m’ is not 1, the inverse modulo does not exist.