Euclidean Algorithm

Purpose: Compute the Greatest Common Divisor (GCD) of two integers, in O(log(min(a, b))) time.

Algorithm

1. Given two integers a and b (assume a ≥ b ≥ 0).
2. If b is 0, the GCD is a — stop.
3. Otherwise, replace (a, b) with (b, a mod b).
4. Repeat until b becomes 0.

Code

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

Paradigm

Decrease and Conquer. Each step reduces the problem to a strictly smaller instance (a mod b is always smaller than b), shrinking toward the trivial base case b = 0 without needing to split into independent subproblems.

Complexity

• Time: O(log(min(a, b)))
• Space: O(1)

Proof of Correctness

Claim: gcd(a, b) = gcd(b, a mod b).

Proof: Let r = a mod b, so a = qb + r for some integer q. Any common divisor d of a and b must also divide a - qb = r, so d is a common divisor of b and r. Conversely, any common divisor d of b and r must also divide qb + r = a, so d is a common divisor of a and b. Since a and b have exactly the same set of common divisors as b and r, they share the same greatest common divisor: gcd(a, b) = gcd(b, r).

Termination: Since r = a mod b always satisfies 0 ≤ r < b, the second element of the pair strictly decreases every step (as long as it’s not already 0), so the sequence of b values is strictly decreasing and bounded below by 0 — it must reach 0 in finitely many steps. When b = 0, gcd(a, 0) = a trivially (every number divides 0, so the greatest common divisor of a and 0 is a itself).

Why O(log(min(a,b))) steps suffice: This follows from a classical result (related to the Fibonacci sequence being the worst case): it can be shown that a mod b < a / 2 whenever b ≤ a/2, and when b > a/2, a mod b = a - b < a/2 directly — either way, after two steps the larger number at least halves. So the number of steps is O(log(min(a,b))). ∎

Variants / Use Cases

• Extended Euclidean Algorithm → additionally finds integers x, y such that ax + by = gcd(a,b), used for modular inverses
• Binary GCD algorithm (Stein’s algorithm) → avoids division/modulo, using only subtraction and bit shifts, useful on hardware where division is slow
• LCM computation → lcm(a,b) = a / gcd(a,b) * b, built directly on this algorithm
• Simplifying fractions → dividing numerator and denominator by their GCD
• RSA and cryptographic key generation → GCD checks are used to verify coprimality of chosen numbers