Compute inverses for singles, lists, or ranges. View Euclidean steps, tables, exports, and a graph. Great for affine cipher lessons and quick class checks.
| Input a | gcd(a, 26) | Inverse mod 26 | Reason |
|---|---|---|---|
| 3 | 1 | 9 | 3 × 9 = 27, and 27 mod 26 = 1 |
| 5 | 1 | 21 | 5 × 21 = 105, and 105 mod 26 = 1 |
| 7 | 1 | 15 | 7 × 15 = 105, and 105 mod 26 = 1 |
| 13 | 13 | None | gcd is not 1, so inverse does not exist |
| 15 | 1 | 7 | 15 × 7 = 105, and 105 mod 26 = 1 |
A modular inverse of a modulo 26 exists only when gcd(a, 26) = 1. If that condition holds, there is an integer x such that:
a × x ≡ 1 (mod 26)
The calculator finds that inverse with the Extended Euclidean Algorithm. It solves:
a × x + 26 × y = 1
Once this equation is solved, the coefficient x becomes the inverse after it is reduced into the range 0 to 25. Since 26 = 2 × 13, any value sharing factor 2 or 13 with 26 has no inverse.
Select a calculation mode first. Use single mode for one integer, list mode for several values, or range mode for a full interval. The modulus stays fixed at 26 because this tool focuses on modulo 26 arithmetic.
Enter your values and choose whether to display Euclidean steps and verification lines. Press the calculate button to show the result block above the form. The output table lists the original value, the normalized residue, the gcd with 26, whether an inverse exists, and the inverse itself.
When an inverse exists, the table also shows a verification statement proving that the product leaves remainder 1 after division by 26. If no inverse exists, the table explains why. The Plotly chart then visualizes invertible inputs and their inverse values.
Use the CSV button to download a spreadsheet-friendly version of the results. Use the PDF button to save a report that includes the summary and the result table. This is helpful for classroom practice, cryptography exercises, affine cipher work, and modular arithmetic checks.
This inverse mod 26 calculator is designed for modular arithmetic, number theory practice, and classical cryptography tasks. In modulo 26 systems, a number has an inverse only when it is coprime with 26. That fact matters in affine cipher decoding, matrix key validation, and modular equation solving.
The calculator handles single values, bulk lists, and ranges, so it works for quick checks and broader pattern analysis. It normalizes negative integers automatically, which means values like -3 are converted into their equivalent residue before testing. This makes the output practical and consistent for real classroom and coding examples.
The built in step output uses the Extended Euclidean Algorithm. That method does more than test invertibility. It also returns the coefficient needed to build the modular inverse. Seeing those steps can help learners understand why some numbers work and others fail.
The result table is structured for review, export, and verification. Each row shows the input, the normalized value, the gcd with 26, whether an inverse exists, and the final inverse when available. A verification line confirms the modular multiplication result.
The graph offers a simple visual of invertible inputs and their inverse outputs. This is useful when you want to scan patterns across a range. Together, the form, table, exports, and graph make the page useful for study, teaching, and lightweight project work.
It is a number b such that a × b leaves remainder 1 after division by 26. The inverse exists only when a and 26 are coprime.
A number has no inverse when it shares a factor with 26. Since 26 equals 2 × 13, every even number and every multiple of 13 fails.
Negative integers represent the same residue classes as certain positive integers. For example, -3 mod 26 becomes 23, so the inverse calculation should use 23.
It uses the Extended Euclidean Algorithm. That method computes gcd(a, 26) and also finds coefficients that produce the modular inverse when the gcd equals 1.
Yes. Affine ciphers require invertible keys under modulo 26 arithmetic. This calculator quickly checks whether a key value is valid and gives its inverse.
Any residue from 0 to 25 with gcd equal to 1 is invertible. Examples include 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, and 25.
They show that input × inverse mod 26 equals 1. This confirms the answer directly and helps you validate the row without extra manual calculation.
Yes. The page includes CSV and PDF download options. They make it easy to save, review, submit, or reuse the computed inverse table.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.