Extended Euclidean Algorithm - Problem

The Extended Euclidean Algorithm is a powerful mathematical tool that not only finds the gcd(a, b) but also determines integer coefficients x and y such that ax + by = gcd(a, b).

This algorithm is fundamental in number theory and has practical applications in cryptography, particularly for computing modular inverses. When gcd(a, m) = 1, the coefficient x gives us the modular inverse of a modulo m.

Task: Given two integers a and b, return a tuple [gcd, x, y] where gcd is the greatest common divisor and x, y are coefficients satisfying ax + by = gcd.

Note: If multiple solutions exist, return any valid solution.

Input & Output

Example 1 — Basic Case

$ Input: a = 30, b = 18

› Output: [6, -1, 2]

💡 Note: gcd(30, 18) = 6, and we can verify: 30×(-1) + 18×2 = -30 + 36 = 6 ✓

Example 2 — Coprime Numbers

$ Input: a = 15, b = 28

› Output: [1, -15, 8]

💡 Note: gcd(15, 28) = 1 (coprime), coefficients satisfy: 15×(-15) + 28×8 = -225 + 224 = -1, but we need |gcd| = 1

Example 3 — One Number is Zero

$ Input: a = 0, b = 12

› Output: [12, 0, 1]

💡 Note: When a = 0, gcd(0, 12) = 12, and 0×0 + 12×1 = 12

Constraints

-10⁹ ≤ a, b ≤ 10⁹
At least one of a, b is non-zero

Visualization

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Asked in

G Google 25 M Microsoft 18 a Amazon 15 f Meta 12

The Extended Euclidean Algorithm elegantly computes both the GCD and Bézout coefficients in one pass. The key insight is tracking coefficient transformations during recursive GCD computation rather than searching separately. Best approach uses recursive extended GCD with Time: O(log min(a,b)), Space: O(log min(a,b)).

Common Approaches

✓ Brute Force Trial

⏱️ Time: O(max(a,b)) Space: O(1)

Test different values of x and y systematically until we find coefficients that satisfy ax + by = gcd(a,b). First compute GCD separately, then search for coefficients.

Extended Euclidean Algorithm

⏱️ Time: O(log min(a,b)) Space: O(log min(a,b))

Use the recursive structure of Euclidean algorithm to simultaneously compute coefficients. At each recursive step, update coefficients based on the division operation.

Brute Force Trial — Algorithm Steps

Compute gcd(a,b) using standard Euclidean algorithm
Try different values of x from -max(a,b) to max(a,b)
For each x, calculate y = (gcd - a*x) / b
Check if y is integer and ax + by equals gcd

Visualization

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Step-by-Step Walkthrough

Compute GCD

First find gcd(a,b) = 2

Try x values

Test x = -3, -2, -1, 0, 1, 2, 3...

Check equation

For each x, verify if ax + by = gcd

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

int gcd(int a, int b) {
    if (a < 0) a = -a;
    if (b < 0) b = -b;
    while (b != 0) {
        int temp = b;
        b = a % b;
        a = temp;
    }
    return a;
}

int max(int a, int b) {
    return (a > b) ? a : b;
}

int abs_val(int x) {
    return (x < 0) ? -x : x;
}

void solution(int a, int b, int result[3]) {
    if (a == 0) {
        result[0] = b; result[1] = 0; result[2] = 1;
        return;
    }
    if (b == 0) {
        result[0] = a; result[1] = 1; result[2] = 0;
        return;
    }
    
    int g = gcd(a, b);
    
    // Try different values of x
    for (int x = -abs_val(max(a, b)); x <= abs_val(max(a, b)); x++) {
        if (b != 0 && (g - a * x) % b == 0) {
            int y = (g - a * x) / b;
            if (a * x + b * y == g) {
                result[0] = g; result[1] = x; result[2] = y;
                return;
            }
        }
    }
    
    result[0] = g; result[1] = 1; result[2] = 0;
}

int main() {
    int a, b;
    scanf("%d", &a);
    scanf("%d", &b);
    
    int result[3];
    solution(a, b, result);
    printf("[%d,%d,%d]\n", result[0], result[1], result[2]);
    
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(max(a,b))

Search through O(max(a,b)) possible values of x

✓ Linear Growth

Space Complexity

O(1)

Only uses constant extra variables

✓ Linear Space

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Extended Euclidean Algorithm - Problem

Input & Output

Constraints

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Related Problems

Common Approaches

Brute Force Trial — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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