Advanced Modular Arithmetic Calculator

Calculator Inputs

Use the fields below to solve modular sums, products, powers, inverses, divisions, and congruence checks.

Operation

Value a

Value b

Modulus n

Exponent e

Compare value c

Graph points

Tip: Addition, subtraction, and multiplication use values a, b, and modulus n.

Quick reference: Normalize each value first, then reduce the result into the residue set {0, 1, 2, ..., n-1}.

Formula Used

Modular arithmetic reduces every number into a residue class determined by the modulus. The calculator uses these formulas:

Addition: r = (a + b) mod n
Subtraction: r = (a - b) mod n
Multiplication: r = (a × b) mod n
Exponentiation: r = a^e mod n
Inverse: a^-1 exists only when gcd(a, n) = 1
Division: a / b mod n = a × b^-1 mod n
Congruence: a ≡ c (mod n) when n divides (a - c)
Normalization: a mod n = ((a % n) + n) % n

How to Use This Calculator

Select the modular operation you want to evaluate.
Enter the main value a, the modulus n, and any extra value needed.
Use b for arithmetic and division, e for powers, and c for congruence tests.
Choose how many graph points you want for the residue pattern.
Click Calculate Now to display the result above the form.
Review the steps, summary table, graph, and sequence table.
Use the export buttons to save the result as CSV or PDF.

Example Data Table

Operation	Inputs	Formula	Result
Addition	a = 17, b = 9, n = 12	(17 + 9) mod 12	2
Subtraction	a = 21, b = 8, n = 7	(21 - 8) mod 7	6
Multiplication	a = 11, b = 4, n = 9	(11 × 4) mod 9	8
Exponentiation	a = 5, e = 4, n = 13	5⁴ mod 13	1
Inverse	a = 7, n = 26	7^-1 mod 26	15
Congruence	a = 38, c = 14, n = 12	38 ≡ 14 (mod 12)	Yes

Frequently Asked Questions

1) What is modular arithmetic?

Modular arithmetic studies remainders after division by a fixed modulus. Instead of keeping full integers, it places values into repeating residue classes such as 0 through n-1.

2) Why are negative numbers normalized?

Normalization converts negative or large integers into standard residues. This keeps answers consistent and makes comparisons easier across all modular operations.

3) When does a modular inverse exist?

A modular inverse exists only when gcd(a, n) = 1. If the number and modulus share a factor, no inverse can satisfy a × a^-1 ≡ 1.

4) What does modular division mean?

Modular division is not ordinary division. It means multiplying by the inverse of the divisor. So a / b mod n becomes a × b^-1 mod n, when the inverse exists.

5) Why does gcd matter here?

The gcd shows whether a value is coprime with the modulus. Coprime values are essential for inverses, reduced residue systems, and multiplicative order.

6) Can the modulus be 1?

Yes, but every integer reduces to 0 modulo 1. That makes the system trivial, and inverses or meaningful residue patterns do not apply.

7) Why use repeated squaring for powers?

Repeated squaring computes large exponents efficiently. It avoids huge intermediate values and significantly reduces the number of multiplication steps needed.

8) What does the graph show?

The graph visualizes how residues cycle under repeated modular operations. It helps reveal patterns, periodicity, and overlap between congruent values.