The Surprising Properties of k Consecutive Integers

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

• A. It is a prime number
• B. It is divisible by k!
• C. It is always an even number
• D. It is equal to k

Answer: (B)

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.

When it comes to understanding k consecutive integers, there’s a rather fascinating mathematical property that you simply can’t overlook: the product of these integers is always divisible by k!. Sounds intriguing, right? Let’s break it down!

Imagine you’re looking at a sequence of consecutive integers, like n, n+1, and n+2, all the way up to n+(k-1). When you multiply these k numbers together—well, you’ve actually packed in every integer factor from 1 to k in that neat little package. This is exactly why their product is divisible by k!, which is the formal way of writing the product of all integers from 1 to k.

Now, let’s take a practical example to illustrate this concept. Say we have k equals 3, so our integers are 2, 3, and 4. The product here is straightforward: 2 x 3 x 4 equals 24. If you take k!, which is 3! (that’s 3 factorial), you get 6. Yeah, 6 divides evenly into 24. You can see how this works! This property applies for any value of k, so whether you choose three integers or fifteen, you can rest assured that the product will always be divisible by k!.

Now, you might wonder about the other options. A product can only be prime if k equals 1—that’s an exception, not a rule. As for it being an even number, that’s not guaranteed unless k is 2 or more; similarly, it doesn’t equal k unless you choose specific integers. For example, if k is 5, the product of 1, 2, 3, 4, and 5 doesn’t just equal 5, it equals 120!

So, when you’re preparing for the GMAT, grasping these concepts can give you an excellent edge in the quantitative section. You’ll find that mathematical understanding isn’t just about memorizing formulas but also about recognizing patterns and relationships like the beauty of divisibility in consecutive integers.

This principle serves as a reminder that the elegant world of mathematics is filled with surprises and connections. Next time you whip out that calculator or pen and pad, remember that the hidden gems within the numbers might just help you solve those tricky problems! Ultimately, understanding the properties of k consecutive integers isn't just an academic exercise; it’s a stepping stone toward mastering essential math concepts and boosting your confidence for the GMAT.