Compute modular addition, multiplication, exponentiation, and inverses. Visualize on clock-face number circles.
Modular Arithmetic Calculator reduces addition, multiplication, exponentiation, and multiplicative inverses to residues for a chosen modulus. Enter the operands required by the selected operation and the modulus; the numerical result is accompanied by a clock-face number circle showing wraparound among residue classes. It answers questions such as where a power lands on a finite cycle. The important trap is the inverse: division modulo n is not ordinary division, and an input has no multiplicative inverse unless it is coprime to n.
A concrete input and expected output from the current implementation.
Input
Operation: exponentiation Base: 3 Exponent: 4 Modulus: 7
Expected output
4
The ordinary power is 3^4 = 81. Since 81 = 7 * 11 + 4, its residue modulo 7 is 4.
For a positive modulus n, addition returns the residue of a + b, multiplication returns the residue of a * b, and exponentiation returns the residue of a^e. Reducing operands before an operation does not change the final congruence. For example, 17 can be replaced by 2 in arithmetic modulo 5 because both leave residue 2.
An inverse request looks for a residue x such that a * x has residue 1 modulo n. Such an x exists exactly when gcd(a, n) = 1. The number circle places residues around one cycle, with 0 following n - 1, so integers separated by a multiple of n occupy the same position.
An inverse of a modulo n must multiply by a to produce residue 1. This is possible exactly when a and n have greatest common divisor 1. For example, 3 has inverse 5 modulo 7 because 3 * 5 = 15 has residue 1. The value 6 has no inverse modulo 9 because their greatest common divisor is 3.
No. It is scoped to modular addition, multiplication, exponentiation, and the inverse of one value. When a is invertible modulo n, a congruence such as a * x having residue b can be solved by multiplying b by that inverse. If a is not invertible, that shortcut is unavailable, and the congruence may have multiple solutions or no solution.
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