Inverse modulo calculator form
Formula used
Main condition: a × x ≡ 1 (mod m)
Existence rule: gcd(a, m) = 1
Extended Euclid form: s × a + t × m = gcd(a, m)
If gcd = 1: the coefficient s is the inverse, reduced modulo m.
How to use this calculator
- Enter the integer
a. - Enter the modulus
m, greater than 1. - Choose graph points and verification rows.
- Press Calculate inverse.
- Read the normalized value, gcd result, steps, and final inverse.
- Use the CSV or PDF button to save the report.
Example data table
| a | m | Normalized a | gcd(a, m) | Inverse | Check |
|---|---|---|---|---|---|
| 3 | 11 | 3 | 1 | 4 | 3 × 4 mod 11 = 1 |
| 10 | 17 | 10 | 1 | 12 | 10 × 12 mod 17 = 1 |
| 7 | 26 | 7 | 1 | 15 | 7 × 15 mod 26 = 1 |
| -3 | 11 | 8 | 1 | 7 | 8 × 7 mod 11 = 1 |
| 14 | 28 | 14 | 14 | No inverse | gcd is not 1 |
FAQs
1) What is an inverse modulo?
An inverse modulo is a number x such that a × x leaves remainder 1 when divided by m. It undoes multiplication inside modular arithmetic.
2) When does a modular inverse exist?
A modular inverse exists only when gcd(a, m) equals 1. If a and m share any larger common factor, no inverse can produce residue 1.
3) Why use the Extended Euclidean Algorithm?
It finds integers s and t satisfying s×a + t×m = gcd(a, m). When the gcd is 1, s becomes the modular inverse after reduction.
4) Can this calculator handle negative a values?
Yes. The calculator first normalizes a into the range 0 to m−1. Then it runs the inverse check and reports the equivalent positive inverse.
5) Is the inverse unique?
Yes, modulo m. All equivalent answers differ by multiples of m, but they represent the same residue class. The calculator shows the smallest nonnegative representative.
6) What if the modulus is prime?
For prime m, every nonzero residue has an inverse. Only values equivalent to zero modulo m fail, because their gcd with m is not 1.
7) Why is the gcd displayed in the result?
The gcd is the decision test. It instantly tells you whether the inverse exists. A gcd of 1 means success; anything larger means failure.
8) Where are modular inverses used?
They appear in number theory, cryptography, coding tasks, modular equations, hashing, and finite-field computations. They are especially common in RSA and algorithm contests.