Graduate Management Admission Test (GMAT) Practice Test

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

1. A perfect square

2. A negative number

3. Prime

4. An even number

The correct answer is: Prime

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.