Understanding the GMAT: What Can’t Be the Product of Two Integers Greater Than 1

If a and b are integers greater than 1, what can their product not be?

• A. A perfect square
• B. A negative number
• C. Prime
• D. An even number

Answer: (C)

When considering the product of two integers greater than 1, it's important to analyze the characteristics of the resulting product based on the properties of its factors. The product of two integers is a prime number only if one of the integers is 1 and the other is itself. However, since both integers a and b in this scenario must be greater than 1, their product cannot be prime. A prime number has exactly two distinct positive divisors: 1 and itself. Since both a and b are greater than 1, their product will have at least four positive divisors: 1, a, b, and their product ab, which confirms that it cannot be prime. On the other hand, the product can indeed be a perfect square if both integers are the same (for example, 2 × 2), a negative number (which cannot be achieved with integers strictly greater than 1 since the product remains positive), or an even number (achievable if at least one of the integers is even). Therefore, the key point is that the combination of two integers, both greater than 1, ensures that their product must always consist of more than two factors, thereby confirming that the product cannot be prime.

When you think about the Graduate Management Admission Test (GMAT), it's easy to get lost in the sea of information thrown at you. You might wonder, "What's really important for my preparation?" Well, let’s break it down with a fascinating math nugget: What can’t be the product of two integers greater than 1?

Can We Talk Numbers?

You know what? When you pick two integers, say a and b, that are both greater than 1, their product starts to take shape. But here's the kicker: it can never be a prime number! That's right. A prime number is defined as one that has exactly two distinct positive divisors—1 and itself. Since both a and b are greater than 1, their product will have at least four factors: 1, a, b, and ab. These will always bring along friends (or factors), validating the idea that primes just can’t hang around when we’re talking about such integers.

Perfect Squares and Other Possibilities

Now, let’s not throw out the idea of perfect squares just yet. Imagine if both integers a and b were the same—like, say, 3 × 3. You get a neat little 9, which is a perfect square. So, yeah, depending on how you play your math cards, the product can indeed be a friendly perfect square.

As for negative numbers? Forget about it! Since you’re only working with integers greater than 1, your product will always be positive. And what about even numbers? Ah, yes! If at least one of those integers is even (like 2), bingo! You can definitely achieve an even product.

So, What’s the Key Takeaway?

In short, the combination of two generally positive integers greater than 1 guarantees that their product will never fit snugly into the prime category. This concept is not just a math exercise; it’s a vital piece of knowledge that can help you tackle GMAT questions with confidence.

Whether you’re practicing problems or simply getting your head around concepts, remember the uniqueness of these properties. The GMAT doesn’t just throw numbers at you for kicks—it’s all about understanding how these integers and their relationships work. If you are prepared to tackle them one by one, you’ll find that math can indeed be friendly!

In the grand scheme, mastering these small but significant nuggets of knowledge can truly redefine your test-taking strategy. And isn't that what every GMAT aspirant is aiming for? Building a solid foundation will not only help you breeze through the GMAT but will prepare you for future challenges in your academic journey. Who knew math could be such a blast and an essential part of your aspirations?