1. **Problem Statement:** Given that $p$ and $q$ are both prime numbers, we want to understand what this implies and explore some properties. 2. **Definition of Prime Numbers:** A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. 3. **Properties:** - Both $p$ and $q$ have exactly two distinct positive divisors: 1 and themselves. - If $p \neq q$, then $p$ and $q$ are distinct primes. 4. **Example:** If $p=2$ and $q=3$, both are prime numbers. 5. **Implications:** - The product $pq$ is a composite number unless $p=q$. - The sum $p+q$ is an integer greater than 2. This is the basic understanding when $p$ and $q$ are prime numbers.