1. The problem is to find the requirements for a four-digit number to be expressed as a product of two prime numbers. 2. Let the four-digit number be $N$, where $1000 \leq N \leq 9999$. 3. Suppose $N = p \times q$, where $p$ and $q$ are prime numbers. 4. Since $p$ and $q$ are primes, both must be greater than 1. 5. To ensure $N$ is four digits, the product $p \times q$ must satisfy: $$1000 \leq p \times q \leq 9999$$ 6. The smallest prime is 2, so the smallest possible product of two primes is $2 \times 2 = 4$, which is less than 1000, so both primes must be larger. 7. To find the range for $p$ and $q$, consider the square root of 1000 and 9999: $$\sqrt{1000} \approx 31.62, \quad \sqrt{9999} \approx 99.995$$ 8. For $p$ and $q$ to produce a four-digit number, both primes should be roughly between 32 and 100, but one could be smaller if the other is larger, as long as their product is in the four-digit range. 9. Therefore, the requirements are: - $p$ and $q$ are prime numbers. - $1000 \leq p \times q \leq 9999$. 10. In summary, a four-digit number is a product of two primes if it can be factored into two prime numbers whose product lies between 1000 and 9999 inclusive.