The Greatest Common Divisor (GCD) & Least Common Multiple (LCM) Calculator determines the greatest shared divisor and lowest common multiple of two positive integers using the classical Euclidean algorithm.

### Mathematical Foundations & Euclidean Algorithm The calculation relies on two fundamental number-theoretic principles: - **Euclidean Algorithm for Greatest Common Divisor**: Computes the GCD of two positive integers $a$ and $b$ ($a \ge b$) by repeated modulo division until remainder reaches zero: $$\gcd(a, b) = \begin{cases} a & \text{if } b = 0 \\ \gcd(b, a \bmod b) & \text{if } b > 0 \end{cases}$$ - **Fundamental Theorem for Least Common Multiple**: The product of two positive integers equals the product of their greatest common divisor and least common multiple: $$a \times b = \gcd(a, b) \times \text{LCM}(a, b)$$ Rearranging gives the exact formula implemented by the tool: $$\text{LCM}(a, b) = \frac{|a \times b|}{\gcd(a, b)}$$

### Worked Number-Theoretic Example Finding the GCD and LCM for $a = 48$ and $b = 72$: - **Step 1: Euclidean Divisor Steps**: - $72 \bmod 48 = 24$ - $48 \bmod 24 = 0$ - Remainder is $0$, terminating the algorithm with $\mathbf{\gcd(48, 72) = 24}$. - **Step 2: Least Common Multiple**: $$\text{LCM}(48, 72) = \frac{48 \times 72}{24} = \frac{3{,}456}{24} = \mathbf{144}$$ - **Verification**: $48 \times 3 = 144$ and $72 \times 2 = 144$, confirming 144 is the smallest shared integer multiple.

### In-Browser Processing & Privacy Note All recursive Euclidean iterations and arithmetic evaluations execute instantaneously inside your client-side browser runtime. No input numbers, math assignments, or calculation logs are sent across external networks.