Least Common Multiple in Java

// Least Common Multiple in Java
//
// PURPOSE
// The least common multiple is the smallest nonnegative common multiple. For positive inputs
// it equals a divided by gcd(a,b), multiplied by b.
//
// METHOD
// Handle zero first. Compute the gcd with Euclid, divide before multiplying to reduce
// intermediate overflow, and return the product.
//
// COMPLEXITY
// O(log(min(a,b))) arithmetic steps, O(1) auxiliary space.
//
// CONTRACT
// Inputs are nonnegative. Returns 0 if either input is zero. The final product must fit the
// integer range.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 36.
// Run: javac AlgorithmExample.java && java AlgorithmExample
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class AlgorithmExample {
    static void alg_require(boolean valid) { if (!valid) throw new IllegalArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int lcm(int a, int b) {
        alg_require(((a >= 0) && (b >= 0)));
        if (((a == 0) || (b == 0))) {
            return 0;
        }
        int x = a;
        int y = b;
        int remainder = 0;
        while ((y != 0)) {
            remainder = (x % y);
            x = y;
            y = remainder;
        }
        return Math.multiplyExact((a / x), b);
    }

    public static void main(String[] args) {
        int a = 12;
        int b = 18;
        int result = lcm(a, b);
        System.out.println(result);
    }
}