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Advanced Modular Functions

This section covers advanced modular functions that handle specific cases such as finding inverses, greatest common divisors, and least common multiples in modular arithmetic.

Functions Overview​

  • modInverse(a: number, mod: number): number Finds the modular multiplicative inverse of a under mod.

  • modGCD(a: number, b: number): number Finds the greatest common divisor (GCD) of two numbers using modular arithmetic.

  • modLCM(a: number, b: number, mod: number): number Computes the least common multiple (LCM) modulo mod.

Detailed Documentation​

modInverse​

Description:​

Finds the modular multiplicative inverse of a under mod, such that:

(a × x) % mod = 1

Example:​

console.log(modInverse(3, 7)); // Output: 5 (since (3 * 5) % 7 = 1)

modGCD​

Description:​

Finds the greatest common divisor (GCD) of two numbers using modular arithmetic.

Example:​

console.log(modGCD(9, 6)); // Output: 3

modLCM​

Description:​

Computes the least common multiple (LCM) modulo mod, such that:

LCM(a, b) = (|a × b|) / GCD(a, b)

Example:​

console.log(modLCM(4, 6, 5)); // Output: 4 (since LCM(4, 6) % 5 = 4)

Error Handling​

  1. Division by zero
  • If a modular inverse does not exist (e.g., when a and mod are not coprime), an error is thrown.
console.log(modInverse(6, 9)); // Throws: "Modular inverse does not exist."
  1. Invalid Inputs:
  • If inputs are not numbers, an error is thrown.
console.log(modGCD("9", 6)); // Throws: "Invalid input: must be numbers."

Advanced use Cases​

Combining with Other Functions​

You can use these advanced modular functions in conjunction with basic operations for complex calculations.

const result = modLCM(modGCD(9, 6), modInverse(3, 7), 10);
console.log(result); // Outputs: Combined result