Final answer:
The product of any set of n consecutive positive integers is always divisible by n! because the set includes multiples or factors of all the integers from 1 to n.
Step-by-step explanation:
The product of any set of n consecutive positive integers is always divisible by n! (option A). To understand why, consider that in any set of n consecutive positive integers, there will be multiples of all the integers from 1 to n. For instance, in a set of 4 consecutive positive integers (like 3, 4, 5, 6), there is at least one multiple of 4 and 2 (which are factors of 4!), and the numbers 1 and 3 themselves are present. Therefore, the product will include all the factors of n! making it divisible by n!.
This also follows from the basic principle that n consecutive integers will have a complete set of remainders when divided by n, meaning one of the integers will be divisible by n, another by n-1, and so on, ensuring that all factors of n! are included in the product.