GCD, LCM & Extended GCD

GCD, LCM, Extended Euclidean algorithm, and linear Diophantine equation solver.

[Algebra]|

Jul 9, 2026

|explanatory_notes.md|

#gcd#lcm#extended-gcd#diophantine#bezout

$cat source_code/

cpp

#include <bits/stdc++.h>
using namespace std;
using ll = long long;

ll gcd(ll a, ll b) {
    while (b) { a %= b; swap(a, b); }
    return a;
}

ll lcm(ll a, ll b) {
    return a / gcd(a, b) * b;
}

ll extgcd(ll a, ll b, ll &x, ll &y) {
    if (!b) { x = 1; y = 0; return a; }
    ll x1, y1;
    ll g = extgcd(b, a % b, x1, y1);
    x = y1;
    y = x1 - (a / b) * y1;
    return g;
}

bool solveLinear(ll a, ll b, ll c, ll &x0, ll &y0, ll &g) {
    g = extgcd(abs(a), abs(b), x0, y0);
    if (c % g) return false;
    x0 *= c / g;
    y0 *= c / g;
    if (a < 0) x0 = -x0;
    if (b < 0) y0 = -y0;
    return true;
}

31 linesutf-8

$cat explanation_notes.md

notes_viewer --renderedmarkdown (math enabled)

GCD, LCM & Extended GCD

Functions

gcd(a, b) — Euclidean algorithm: replace

(a, b)

with

(b, a \bmod b)

until

b = 0

lcm(a, b) —

\text{lcm}(a, b) = \frac{a}{\gcd(a,b)} \cdot b

(divide first to avoid overflow)

extgcd(a, b, x, y) — finds

x, y

such that

ax + by = \gcd(a, b)

(Bezout's identity)

solveLinear(a, b, c, x0, y0, g) — solves

ax + by = c

. Solution exists iff

\gcd(a,b) \mid c

General Solution for $ax + by = c$

Given one solution $(x_0, y_0)$ , all solutions:

x = x_0 + \frac{b}{g} \cdot t, \quad y = y_0 - \frac{a}{g} \cdot t

for any integer $t$ .

When to Use

Fraction simplification, divisibility checks

Modular inverse when mod isn't prime

Solving Diophantine equations, CRT steps

Complexity

All functions —

O(\log \min(a, b))

GCD, LCM & Extended GCD

GCD, LCM & Extended GCD

Functions

General Solution for ax+by=cax + by = cax+by=c

When to Use

Complexity

General Solution for $ax + by = c$