1. The problem is to understand what the greatest common divisor (GCD) is. 2. The GCD of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. 3. For example, to find the GCD of two numbers $a$ and $b$, we look for the biggest number $d$ such that $d \mid a$ and $d \mid b$ (meaning $d$ divides both $a$ and $b$). 4. One common method to find the GCD is the Euclidean algorithm, which uses repeated division: $$\text{gcd}(a,b) = \text{gcd}(b, a \bmod b)$$ until $b=0$, then the GCD is $a$. 5. For example, to find $\text{gcd}(48,18)$: - Compute $48 \bmod 18 = 12$ - Then $\text{gcd}(48,18) = \text{gcd}(18,12)$ - Next, $18 \bmod 12 = 6$ - So $\text{gcd}(18,12) = \text{gcd}(12,6)$ - Then, $12 \bmod 6 = 0$ - So $\text{gcd}(12,6) = 6$ 6. Therefore, $\text{gcd}(48,18) = 6$. 7. The GCD is useful in simplifying fractions, finding least common multiples, and solving problems in number theory.