1. The problem is to understand and apply the Euclidean algorithm to find the greatest common divisor (GCD) of two integers. 2. The Euclidean algorithm is based on the principle that the GCD of two numbers also divides their difference. The formula used is: $$\gcd(a,b) = \gcd(b, a \bmod b)$$ where $a \bmod b$ is the remainder when $a$ is divided by $b$. 3. Steps to apply the Euclidean algorithm: - Start with two positive integers $a$ and $b$ where $a > b$. - Divide $a$ by $b$ and find the remainder $r$. - Replace $a$ with $b$ and $b$ with $r$. - Repeat the process until $b$ becomes 0. - The GCD is the last non-zero remainder. 4. Example: Find $\gcd(48, 18)$. - $48 \div 18 = 2$ remainder $12$, so $\gcd(48,18) = \gcd(18,12)$. - $18 \div 12 = 1$ remainder $6$, so $\gcd(18,12) = \gcd(12,6)$. - $12 \div 6 = 2$ remainder $0$, so $\gcd(12,6) = 6$. 5. Therefore, $\gcd(48,18) = 6$. This method efficiently finds the greatest common divisor by repeated division and remainder operations.