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Factors are numbers that divide another number evenly, leaving no remainder. For example, the factors of the number 6 are 1, 2, 3, and 6. Factors are important because they help us understand the structure and properties of numbers. When working with prime numbers, the situation is quite simple. Each prime number has exactly two factors: 1 and the number itself. For instance, consider the number 5. It is prime, and its only factors are 1 and 5. This uniqueness characteristic makes prime numbers the building blocks in the realm of numbers. For composite numbers, which we'll talk about next, the process of finding factors involves discovering more than just the number and 1. Understanding how to find factors is crucial when determining the number of ways a number can be divided into whole numbers.

Composite numbers are numbers that have more than two factors. They are opposite to prime numbers. While a prime number has only 1 and itself as factors, composite numbers have additional divisors. For example, the number 12 has factors 1, 2, 3, 4, 6, and 12. In the context of the problem given, when you multiply three distinct prime numbers, you generate a composite number. For instance, if the distinct prime numbers were 2, 3, and 5, their product is 30, and 30 is composite with factors beyond the prime numbers itself. By multiplying primes, you form new numbers that can be factored further into a variety of combinations, but all these combinations include the original primes. Identifying whether a number is composite helps determine its divisibility and the richness of its factor structure.

Distinct prime numbers are prime numbers that are different from each other. For example, consider 2, 3, and 5. They are all prime and distinct. When we say numbers are prime, we mean that each can only be divided by 1 and itself. In a problem involving distinct prime numbers, like multiplying them, the significance lies in how these bases develop new factors through multiplication. For any set of distinct prime numbers, multiplying them results in a composite number with specific properties. Such a product would have factors including all the original prime numbers and combinations of them, like the product of any two or all three of them (i.e., their combinations such as 1, a, b, c, ab, ac, bc, abc). Understanding the uniqueness and characteristics of distinct prime numbers helps in calculating factors and demonstrates how primes construct more complex number systems.