GMAT Instructor
Joined: 01 Jul 2017
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Concentration: Leadership, Organizational Behavior
Re: If n is an integer, what is the greatest common divisor of 12 and n ?
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06 Oct 2018, 07:04
Don't fall for the trap of this question. It isn't about the math. Many explanations of Quantitative questions focus blindly on the math, but remember: the GMAT is a critical-thinking test. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time (giving you more time for harder questions.) The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. Ready? Let's talk strategy here. Here is the full "GMAT Jujitsu" for this question:
The first thing we need to understand is the concept of a greatest common divisor (often called the greatest common factor or \(GCF\).) The GCF is the largest factor that is shared between two integers. And yet, once we know this, the fundamental trap of this problem is getting you to think that you actually need to solve for the GCF, instead of stopping as soon as you know you CAN solve. Many people spend too much time on Data Sufficiency questions because they think they need to get to the bitter end. We don’t. As soon as we have enough information to conclude that a statement is either sufficient or insufficient, we can move on.
For example, let’s evaluate statement #1. It states that the “product of \(12\) and \(n\) is \(432\).” Immediately, you should recognize that you can solve for \(n\), since \(12n = 432\). One equation. One variable. No weirdness. No possibility of multiple values. You don’t have to do the long division here -- actually solving for n -- to determine that statement #1 is sufficient. After all, if you can solve for \(n\), you can obviously determine what the factors of \(n\) are, and thereby determine what the GCF of \(12\) and \(n\) are. Case closed. Statement #1 is sufficient. Notice that you don't even need to know the answer to the question "what is the greatest common divisor?"
Statement #2 is similarly deceptive, but in a different way. Here, you can’t solve for a specific value of \(n\). However, if the “greatest common factor of \(24\) and \(n\) is \(12\)”, then \(12\) is clearly a factor of \(n\). And we also know that \(12\) is the greatest factor of \(12\) (after all, you can’t have an integer factor of a number greater than the number itself.) So, no matter how you look at it, we know that the greatest common factor of \(n\) and \(12\) must be \(12\). Statement #2 is also sufficient.
With both the statements individually sufficient, the answer is D.
Now, let’s look back at this problem from the perspective of strategy. This problem can teach us several patterns seen throughout the GMAT. First, minimize your math. I have met many engineers who think the Quantitative portion of the test is all about the math, and they end up doing pretty poorly on the GMAT because they fail to realize the test is actually measuring something besides their math skills: critical-thinking. This is especially true with Data Sufficiency. You only need to do "sufficient" math to prove that there must be only one answer to the question. (And this problem requires almost zero math if you think about it.) Second, statements in Data Sufficiency questions often bait you into thinking that you solve each statement in the same way. But the GMAT rewards flexible-thinking, not linear-thinking. With Statement #1, we can solve for \(n\). With Statement #2, it is impossible to know what \(n\) is. But the question isn't asking what \(n\) is. It is basically asking, "is there enough information here to calculate the GCF of two numbers?" On GMAT questions, you must always focus on exactly what the actual question is asking. For this question, we can arrive at an answer by leveraging the information in the problem. And that is how you think like the GMAT.