Chapter 1: Understanding Composite Numbers

To provide a brief overview, a composite number is defined as any number that is not prime, meaning it has factors besides just 1 and itself. In this exploration, our goal is to identify a sequence of 2024 composite numbers. Given the size of 2024, we can infer that the solution will require more than mere guesswork!

Best of luck, and continue reading for the solution.

Section 1.1: The Key Insight

A crucial point to note is that for any integer n, if a number k is divisible by n, then k + n will also be divisible by n. For instance, since 24 is divisible by 6, it follows that 24 + 6 = 30 is also divisible by 6. While this principle is straightforward, applying it to solve our problem requires deeper thought.

To achieve our goal of finding 2024 consecutive composite numbers, we need to find a number that possesses exactly 2024 factors. Each of these factors can be added to our chosen number, resulting in 2024 composite numbers. Additionally, these numbers need to be consecutive, which necessitates that the factors themselves are also consecutive.

To simplify our task, let’s find a number that is divisible by all integers from 2 to 2025. The number that meets this criterion is 2025!, which represents the factorial of 2025. This factorial is the product of all integers from 1 to 2025. Because 2025! is derived from the multiplication of all these integers, we can be assured they are all factors.

Subsection 1.1.1: Generating Composite Numbers

Using our earlier observation, we can add 2 to 2025! and still maintain a multiple of 2. Similarly, adding 3 will yield a multiple of 3, and this pattern continues until we add 2025 to 2025!, resulting in 2024 consecutive composite numbers. These numbers take the following form: