Work with modular residues confidently today. Find inverses, validate gcd rules, and inspect normalized classes. Get accurate inverse residue answers with useful verification details.
| Residue a | Modulus m | gcd(a, m) | Inverse | Check |
|---|---|---|---|---|
| 3 | 7 | 1 | 5 | 3 × 5 ≡ 1 (mod 7) |
| 5 | 12 | 1 | 5 | 5 × 5 ≡ 1 (mod 12) |
| 8 | 15 | 1 | 2 | 8 × 2 ≡ 1 (mod 15) |
| 11 | 26 | 1 | 19 | 11 × 19 ≡ 1 (mod 26) |
| 6 | 15 | 3 | No inverse | Not coprime |
The calculator solves the congruence a × x ≡ 1 (mod m).
An inverse exists only when gcd(a, m) = 1.
It uses the extended Euclidean algorithm to find integers x and y such that a × x + m × y = 1.
The coefficient x becomes the modular inverse. The final answer is reduced into the chosen residue class.
An inverse residue class calculator helps you solve congruences faster. It is useful in number theory, algebra, coding, and cryptography. A modular inverse is the value that turns a residue into one under modular multiplication. If a × x ≡ 1 mod m, then x is the inverse of a modulo m.
The key rule is simple. An inverse exists only when a and m are coprime. That means their greatest common divisor must equal one. If the gcd is greater than one, the inverse does not exist in that residue system. This calculator checks that condition before showing any inverse output.
First, it normalizes the input. Negative values and large values are reduced into a standard residue class. Next, it runs the extended Euclidean algorithm. This method finds coefficients that satisfy Bézout’s identity. From that identity, the inverse can be read directly and reduced to the least positive or symmetric representative.
The output gives more than one number. It shows the normalized residue, the modulus, the gcd, the inverse, and a verification line. It also shows the full inverse class. That class contains every number congruent to the inverse modulo m. This is helpful when you want a complete algebraic description instead of one representative.
This inverse residue class calculator is also good for practice. Students can compare manual work with the displayed Euclidean steps. Teachers can use the example table to explain congruence, residue systems, and multiplicative inverses. Engineers and programmers can confirm modular arithmetic quickly when building formulas, validation rules, or encryption logic.
Use this page when you need a reliable modular inverse answer with proof details. It is ideal for homework, exam revision, algorithm testing, and math demonstrations. Because it verifies each answer, it also reduces mistakes. That makes the calculator practical for both beginners and advanced users working with inverse residue classes.
It is the residue class that multiplies with another residue class to produce 1 modulo m. In practice, people often refer to its least positive representative as the modular inverse.
A modular inverse exists only when the residue and modulus are coprime. That means their greatest common divisor must be exactly 1.
Normalization converts negative or large integers into their standard residue modulo m. This keeps the inverse calculation consistent and easier to interpret.
No inverse exists in that modulus. The calculator reports this clearly because the congruence a × x ≡ 1 (mod m) cannot be satisfied.
The least positive inverse is between 0 and m−1. The symmetric inverse may be negative and is chosen to sit closer to zero.
The steps show how the gcd and inverse were derived. This is useful for learning, checking manual work, and understanding Bézout’s identity.
Yes. The calculator reduces the entered value modulo m first. Then it computes the inverse of the normalized residue if one exists.
It is useful in modular arithmetic, cryptography, coding theory, linear congruences, and classroom math practice where inverse residue classes appear often.