Graduate Management Admission Test (GMAT) Practice Test

For k consecutive integers, what is the mathematical property of their product?

1. It is a prime number

2. It is divisible by k!

3. It is always an even number

4. It is equal to k

The correct answer is: It is divisible by k!

The product of k consecutive integers exhibits the property of being divisible by k!. This is because when you multiply k consecutive integers, the result encompasses all the integer factors from 1 to k. To illustrate, consider any k consecutive integers, such as n, n+1, n+2, ..., n+(k-1). When we calculate the product of these integers, their combined multiplication naturally includes every integer up to k. Since k! (k factorial) is defined as the product of all integers from 1 to k, it follows that this product includes all necessary factors for divisibility by k!. For example, if k equals 3, the product of three consecutive integers (say 2, 3, and 4) is 24, and 3! equals 6, which divides 24 evenly. This holds true for any value of k, affirming that the product of k consecutive integers will always be divisible by k!. Other potential options do not apply universally across all k consecutive integers: a product can be prime only if k equals 1, it may not always be even, and it certainly doesn't equal k itself unless k is specifically chosen to fit those integers, which doesn't hold across the board.