Data Sufficiency Strategies

Back in the old days the GMATPrep tests did not break down your performance by question type on the Quant and Verbal sections. Nor was there an Enhanced Score Report that did so on the actual GMAT. Now, with the existence of practice tests that indicate your performance on each individual question type and the Enhanced Score Report that does the same for the actual test, it is possible to understand how your overall score is arrived at and what your score is for each specific question type.

With that, I have noticed a trend with my students that I could not have known in the past and that probably has relevance for many GMATers out there: they tend to score higher on Data Sufficiency than on Problem Solving. Now, I don’t mean to imply that everyone scores higher on DS than PS – people who don’t really understand Data Sufficiency tend to underperform there – predictably. But what I would say and what I will explain below is this: for people who are not naturally great at Math, there is an opportunity to really excel at Data Sufficiency and have that score pull up the entire Quantitative score.

First, let me explain a little bit about the scoring. When people take practice tests on the official software or when the look at their Enhanced Score Report (ESR), what they tend to look at are the percentiles. This is especially true on the ESR since the percentiles are what appear on the center of the page, but if you look at the bottom of each page in the “Summary” section you will see hidden down there a breakdown of what your score was for each question type, and this information is what is most telling. Your overall score tends to be roughly the average of your scores for each question type. So if you scored 40 on Problem Solving and 48 on Data Sufficiency, your Quant score will probably be 44.

What I started to notice a while back in reviewing practice tests and ESRs with my students is that they usually do better on DS, sometimes overwhelmingly so. Now, they don’t necessarily come to me in that state – often it is the opposite on the early practice tests that they take and DS is much worse. But as we start to approach their test date and they are ramping up and taking a lot of practice tests, most of my students just score higher on DS. A very common scenario for me is to see something like 49DS/42PS for a Quant score of 45 or 46. And at the end of the day they may end up at 49DS/45PS for a 47 Quant.

Here is what is significant about that. Many of these students are not great at Math and are probably not capable of getting really hard Problem Solving questions right on a consistent basis. But the beauty of Data Sufficiency is that it is much more about logic and reasoning than it is about Math. So if you really understand the question type well you can dominate on DS without really being master of the Math that underlies the questions. And what I have found with my students is that it is much easier to get them to a place where they are killing it on DS than on PS, in part because PS questions, especially at the higher levels, really do require a little bit more Math know-how or even innate Math ability.

It is outside the scope of this post to really get into the specifics of how to dominate on Data Sufficiency (though many of my other posts expound upon some of those strategies). And obviously if you are aiming for a 49 Quant, you need to be equally dominant on PS questions. That said, I would argue that if you are not naturally good at Math, Data Sufficiency really presents an opportunity to far outperform your natural Math ability and end up with a score that puts you above those who are innately better at Math!

I tutor plenty of people who routinely get scores of around 40 on PS and in my opinion are unlikely to ever be hitting scores like 44 and above (they are just not that great at Math and have come a VERY long way to even get to a 40 on PS). Yet they will routinely get scores of 47 to 49 on DS. So they come away with scores of 44 or 45 on Quant. Again, for someone aiming for a 49 or 50 Quant, that might not seem that impressive, but for people starting at 30 Quant who are not very good at Math, a 44 or 45 Quant is pretty damn impressive! And for the people I tutor who are able to get to 44 or 45 or 46 on PS (again often people who are not great at Math), they will often end up with a 47 or even 48 Quant, in part because of the higher DS score.

So if you feel like you are the type of person who is just innately not that good at Math, consider that, in Data Sufficiency, you have an opportunity to really dial into the logic of the question type (along with its common structures, traps, etc.) in a way that would allow you to far exceed what you might expect from your pure Math ability.

In a previous post I discussed the tendency that people have to “go on a hunch” on Data Sufficiency questions. This is a big no-no because DS questions are designed to punish people for making unwarranted assumptions. Generally speaking it is best to try to prove what you think to be true on Data Sufficiency. Nevertheless, there are some cases in which it would not be wise to go that far, cases in which it might be better to make a good educated guess rather than spend a lot of time proving out your hunches.

The Problem That Most People Have: Not Pushing the Statements Far Enough

The problem that most people have on Data Sufficiency is that they don’t realize how far they need to go to prove the statements to be sufficient or not sufficient. Test takers are often satisfied with thinking that a statement is sufficient or not sufficient even though they are not really that sure. But as I often tell the people that I tutor, when you are 80% sure on Data Sufficiency that really means that you have about a 50% chance of getting the question right. That is because the statements are designed to appear one way when in fact they are the opposite (i.e, they appear to be sufficient when in fact they are not). So generally speaking you should push the statements further until you reach near 100% certainty on your judgment (see my post about “going on a hunch” for more about how to “prove” the statements are what you think they are).

The Crucial Decision Point

Yet there are cases in which it might be better to not spend the time to reach that higher level of certainty. Essentially there are 2 factors that you need to consider when you are evaluating a statement and deciding how far to go. You need to weigh your level of certainty against the amount of time that it would take to reach a higher level of certainty. For example, if you are close to 100% certain that a statement is sufficient and if it would take another 2 minutes to prove that it is sufficient, it would probably be best to take the educated guess that you have and not spend the extra 2 minutes. If, on the other hand, you are 75% sure and it would only take an additional 30 seconds to be more certain, that is an easy decision: spend the extra 30 seconds.

An Example

Let me give some examples of both sides of that decision process. I am going to use a question that I used before in another post because it so perfectly illustrates how this decision making process should unfold. Consider the first example:

In the above question, the statements don’t appear to be sufficient individually at first glance. Statement 2 definitely cannot be sufficient because without knowing what k is there would be no way to know what the remainder is since essentially we would be free to add whatever we want to 3^22 and could therefore change the remainder. Therefore we can say with certainty that statement 2 cannot be sufficient. Most people look at statement 1 and think the same thing: if we don’t know anything about n, we cannot answer the question. There are 2 things, however, that should give you pause. First, n is related to the exponent and is not just a number that we are adding on to the 3^(4n+2) so it could be that there is some pattern in what results when you take 3 to different exponents. The other thing is that if you conclude that statement 1 is also not sufficient, you would end up at answer choice C, which seems suspiciously obvious in this case. Of course if we know the value of n and k we will be able to answer the question. That is almost certainly too straightforward for a Data Sufficiency question.

So at this point you would find yourself at an important decision point, one that presents itself on many DS questions. The answer could possibly be C, though that is unlikely. The only plausible alternative, given that we know that statement 2 is definitely not sufficient, is that the answer could be A. Now, here is where you need to think about the amount of time it would take to prove that statement 1 is sufficient or not and weigh that against how certain you are about what the answer is on this question. If you were really unsure and if you thought you could work out very quickly whether there is a pattern with statement 1 such that it would be sufficient, then it might be worth doing that. However, if you feel really confident that the answer cannot be C and that choice A is the only plausible alternative and if you further realize that it will take you a very long time to prove that statement 1 is sufficient, it would probably be better to just guess A and move on. The answer in the end is A.

Another Example

Lets take a look at another example, one that will illustrate the other way that this decision process can unfold. Again this is a question I have used in other posts:

Tom, Jane, and Sue each purchased a new house. The average (arithmetic mean) price of the three houses was $120,000. What was the median price of the three houses?
(1) The price of Tom’s house was $110,000
(2) The price of Jane’s house was $120,000

On this question most people blow through statements 1 and 2, believing them each to be not sufficient and then come to choice C. But again, just like the last question, choice C seems deceptively easy. Of course knowing 2 of the home prices and the average would be enough to calculate the third and therefore the median. Again we face this all important decision point. How certain are we and how long would it take to achieve a higher level of certainty? Well in this case it really doesn’t take too long to test out numbers and see if we can indeed get more than 1 median. On statement 1 we can, and so most people then assume that statement 2 is also not sufficient since it seems to provide the exact same type of evidence as statement 1.

But again, how certain are we and how long would it take to become more certain? Most people are pretty sure in this case that statement 2 is not sufficient, but again it would not take very long to test numbers to see if we can get 2 different medians. If it takes an extra 30 to 45 seconds to be completely certain, that is probably worth it. Consider that if you don’t spend that extra 30-45 seconds and get the question wrong then you have wasted much more than 30-45 seconds. So again, if you are not close to 100% certain and if it wouldn’t take that long to be more certain, it is worth spending the time. In this case you would be rewarded for pushing statement 2 because it actually is sufficient, much to most people’s surprise.

Conclusions

To summarize, it is important on the one hand to push the statements and try to arrive at an answer with a very high degree of certainty. “Thinking” that a statement is sufficient or not sufficient is just not enough on Data Sufficiency since things so often turn out to be contrary to expectation. That said, there is an important decision point that comes on every DS question when you must ask yourself, “how certain am I and how much time would it take to become significantly more certain?” If you are almost 100% certain and achieving a higher level of certainty would take a long time, it is probably best to go with your gut and move on. But if you are not nearly as certain and if it would not take that much time to achieve a higher level of certainty then you should probably spend the time to really prove what you think to be true.

There are many things that tend to give people trouble on Data Sufficiency, but one of the most common issues that people face is the problem of “information overload.” One of the skills that Data Sufficiency tests is a person’s ability to simultaneously juggle multiple pieces of information. If you forget that the question restricted you to positive integers you will probably get the question wrong. Lose sight of what your goal is as you are plugging in numbers and you will probably get the question wrong.

So how do you make sure that you take stock of all of the information without getting completely overloaded in the process? There are several strategies that can help you be more organized and focused on Data Sufficiency.

First of all, you must slow down and take your time with the question stem. Most Data Sufficiency neophytes blow through the question stem in their rush to get to the statements. Don’t be a Data Sufficiency neophyte!!! The question stem often provides crucial information, both in terms of restrictions imposed and facts that are pertinent to the question being asked. If you don’t take proper stock of this information you are heading into the statements with a severe handicap.

In terms of the restrictions it might be helpful to jot down them down on your notepad so that they are staring you in the face as you work on the problem. Something as simple as:

Restrictions:
X is a positive integer
Y < 0

And in terms of the other factual information provided in the question stem, you need to try to “digest” it so that you can more easily understand what it is saying. This is often the crucial step on many DS questions. So for example, consider the following question stem:

If x is greater than 0 and if 5x + 6 < 3x + 16, is x a multiple of 3?

In the above case, you MUST process the information given before turning to the statements. If you solve for x in the inequality, you get x < 5. This is a critical step because the question already limits you to values of x that are greater than 0, so essentially x must be between 0 and 5. The only multiple of 3 in that range is 3, so basically the question becomes, is x = 3? I wouldn’t say that you can ‘t answer this question correctly if you don’t take these initial steps, but you are certainly making your job much, much harder.

The other major strategy that I would argue is crucial to avoiding information overload, especially on questions that involve plugging in numbers (i.e., number property questions), is to be goal oriented when turning to the statements so that you have a good sense of what the point is of what you are about to do. That may sound obvious, but most people don’t do it. In fact, even when I am tutoring someone and explain this process and then give a student a question with which to practice the technique, they usually still don’t do it!!!

I could write a whole post or even several posts about just this technique, but for now let me try to keep it somewhat general and perhaps give one example to illustrate.

Before turning to the statements, you should have some idea of how you are going to deal with them and ready yourself accordingly. This would require you to glance at the statements to understand what type of information they provide (equations, facts in sentence format, etc.). This doesn’t mean that you should actually start working on the statements but you should get an idea of the type of information they provide so that you can know what your goal might be and how you are going to deal with the statements.

Again, this is probably most important on number property questions. Take the following question for example:

If m is an integer, is m odd?

(1) m/2 is not an even integer

(2) m – 3 is an even integer

There are multiple ways to approach this question, but especially with statement 1, picking numbers is a very good option. However, many people get spun around when dealing with statement 1, in part because they are not clear about what their goal is before turning to the statements. It’s important, first of all, to understand that the issue on all DS questions is that you are trying to determine if there is one answer or more than one answer to the question given. If there is one answer, the statement is sufficient; if there is more than one answer it is not sufficient.

So if you pick numbers that satisfy a particular statement, you are going to get an answer – that doesn’t really tell you anything since there is always at least one answer to every Data Sufficiency question. The real question is whether there is more than one answer possible to the question given.

So in the above problem, since the question is asking if m is odd, the goal is actually to see if m can be both odd and even (since m must be an integer, m must be either odd or even – it can’t be a non-integer). Again this may seem obvious to some, but so many people fail to fully appreciate that this is the goal on the above question. It bares repeating: The goal is to see if, given the statements, m can be both odd and even. This simple step allows one to stay focused on what the goal is when things start to get complicated at the statements – without it many people will suffer from information overload and get completely spun around, forgetting what the point of what they are doing actually is.

So again in the above example, when dealing with statement 1, the goal would be to satisfy the statement with both an odd and even value of m. If we plug in m=3 we would see that 3/2 is not an even integer so we satisfied the statement with an odd value of m. Now, if we plug in m=4 (in an effort to try an even value of m) we will see that it doesn’t work: 4/2 equals 2 so that is an even integer and that violates the statement. But that doesn’t mean that we should just stop there. We already got our odd value of m, but just because the first even value we tried didn’t work does not necessarily mean that no even value would work.

Here again is where having a goal in mind helps. I have seen countless people get spun around at this point and either start plugging in odd values, which is pointless since we already proved that m can be odd, or sort of flail about and not know what to do next. Our goal was to see if we could get both an odd and even value of m to work in the statements. We got our odd value so all we want to try at this point is even values. M=4 did not work, so we want to ask, “is there an even value for m that I can plug in that would make m/2 not an even integer?” Well 2 is an even integer and 2/2 equals 1 and that is not an even integer, so that satisfies statement 1. So m could also equal 2. That is our even integer. Statement 1 is therefore not sufficient since m could equal an odd or an even.

We can do a similar thing with statement 2. If we make m = 7, then 7-3 equals 4, which is even, so we succeeded in getting an odd value of m to satisfy the statement. Again people get confused here because the result of putting 7 in for m is even, but as long as we remember that our goal was to plug in an odd value for m and have it satisfy the statement, we shouldn’t get confused.

If we try to get an even value to work in statement 2, however, we will not be able to do it. If we try m = 8, 8 – 3 is odd so that violates the statement. If we try m = 10, we will get another odd result. We can keep trying but at a certain point it will become obvious that every even value that we try for m is going to yield an odd when we subtract 3. So we are only able to get an odd value of m to work in statement 2 and therefore statement 2 is sufficient. So the answer to the question is B.

Again there are other ways to deal with this question and for some people the process of picking numbers described above would be obvious and intuitive, but I have seen many, many people struggle with this question and ones like it. One of the keys to not getting lost in questions like these is to be goal oriented and to know what that goal is as you head into the statements. Without that focus many people get overwhelmed by the combination of info in the statements and question stem itself (on this question all of the mention of odd and even in both places often overwhelms and confuses people).

So on Data Sufficiency questions take your time with the question stem, process all of the information given to you, take stock of any restrictions that are imposed, and know explicitly what your goal is before turning to the statements!

When I tutor I often like to ask my students, “what makes a hard question hard from the point of view of the test writers?” Invariably I get answers that point to specific techniques that are used to dial up the level of difficulty, such as abstraction, complicated Math, etc. All true. However, at a more fundamental level hard questions are designated hard because most people get them wrong. If most people got them right then they would, by definition, be easy questions. This leads to a very, very important fact about GMAT questions: when the answer that you are about to choose seems really obvious, the question is either super easy or you are falling into a trap.

This is applicable to all GMAT question types but it is perhaps most important on Data Sufficiency questions. If you take a look in the Official Guide you will see that even on easy questions the obvious answer is usually wrong. In a previous post I wrote about the “Spectrum of Sufficiency” and made a similar point, but I want to take this one step further and describe a concept that I call “bowling with the bumpers in the gutters on Data Sufficiency.” If you have ever gone bowling with little kids you will be familiar with the bumpers that they put in the gutters to prevent the ball from falling in. Well, being aware that obvious answers are usually wrong on DS questions is kind of like bowling with the bumpers in the gutters: when you are about to fall into the gutter and pick the wrong answer you get “bumpered” back into the question.

This is powerful stuff – I personally feel very confident on DS questions in part because I know the test so well that if I happen to make a mistake and start to fall into a trap on a DS question I usually get bumpered back into the question. So it is almost difficult for me to get a DS question wrong because even when I make a conceptual or calculation error I often find myself going back to the question to reconsider – the same thing does not really happen on Problem Solving questions or at least not as often.

Let me illustrate with an example. Consider OG Data Sufficiency Question #110:

#110 Tom, Jane, and Sue each purchased a new house. The average (arithmetic mean) price of the three houses was $120,000. What was the median price of the three houses?

(1) The price of Tom’s house was $110,000

(2) The price of Jane’s house was $120,000

Now, most people spend some time on statement 1 and realize, correctly, that it is not sufficient. At this point most people blow through statement 2 and infer, incorrectly, that since it seems to provide the same kind of information as statement 1 it must also be not sufficient. That would appear to be a reasonable conclusion. But that conclusion would then lead a person to answer choice C. Most people just choose C and never look back, but anyone with a good understanding of Data Sufficiency would probably be suspicious of choice C – and with good reason. Obviously if we know the average and we know the price of 2 of the homes then we would be able to figure out the price of the third home and therefore the median.

If you were about to choose choice C, two thoughts should occur to you at this point. One, what would be the point of this question if the answer was C? There would be no clever thinking, no difficult reasoning required to come to the answer – that is very unGMAT-like. Second, would almost all people come to that same answer with the same general ease? If so the question is either really, really easy or choice C is not the right answer. I wouldn’t say that you should completely write of choice C and absolutely not pick it, but you should at least get “bumpered” back into the question and consider your options again (and here is where the “Spectrum of Sufficiency” can be helpful so please see my previous post if you haven’t already). If you really proved that statement 1 is not sufficient then you really only have one alternative (given that together the statements are definitely sufficient). You need to consider statement 2 again. Doing so and knowing in advance that it probably is sufficient (because choice C is just too easy) is very powerful since you are analyzing it with the foreknowledge of what the correct answer is already likely to be. This is the power of bowling with the bumpers in the gutters on Data Sufficiency.

Some people worry that they will not be able to tell what is truly “too obvious” to be correct. Indeed it takes some practice and most of all some repetition with DS to know what would typically be a trap answer. But again, it doesn’t mean that you absolutely cannot pick the “obvious” answer. It just means that you should question and further analyze things that seem too obvious on Data Sufficiency questions.

Remember, in order to have a high Quant score you need to be getting hard question right. But again, a hard question is hard because most people get it wrong. So in general on the GMAT and especially on Data Sufficiency the right answer on a hard question is not the answer that most people will pick. So if you are about to choose such an “obvious” answer consider that you may be about to fall into the trap that most test takers fall into.

GMATers, especially newcomers to the test, tend to spend on average less time on Data Sufficiency questions than Problem Solving ones. This is in part because Data Sufficiency questions don’t require an “answer” in the same way that Problem Solving questions do, so it is a little harder to achieve the same level of certainty when doing a DS question. Test takers, therefore, often “go on a hunch” and select an answer that they “think” is right or that they are “pretty sure” is right. This is a big no-no! As this post will explain, guessing and knowing that you are guessing on DS questions is fine, but selecting an answer because you are “pretty sure” it is right (and believing that you will most of the time be right) is a great way to get a lot of Data Sufficiency questions wrong.

The first thing that you need to understand is that one of the prime features of Data Sufficiency questions, perhaps even their raison d’etre, is to lure people into making false assumptions. In other words, the questions are designed with that very purpose in mind, because one of the things that Data Sufficiency tests is the willingness of a person to rest on unproven assumptions vs their desire to try to draw conclusions based on proven facts. Obviously in the real world, business decisions should be made based on real evidence and should be arrived at with as high a level of certainty as possible, so this is part of what the GMAT is getting at through its use of Data Sufficiency. With this in mind, DS questions are often designed so that what SEEMS to be the case is often not the case and the reward is there for the person who attempts to prove out his or her hunches.

In my opinion (and this is something I will elaborate on in a future post), there are 3 main ways that you can evaluate the statements on a Data Sufficiency question. You can take a conceptual approach, an algebraic approach, or a picking numbers approach. The approach that you take depends of course on the question itself and it should also be noted that in some cases you may use a combination of these approaches and in other cases perhaps something else entirely. But the point is that many people are comfortable taking a conceptual approach when they really shouldn’t be. When I am tutoring someone, I will often hear, “I am pretty sure statement 2 is sufficient,” to which I will usually respond, “can you prove that” or “what is your level of certainty?”

This latter question is important because if you are 99% certain, then it is ok to stay conceptual since at that point you are not really assuming – conceptually you really understand why the statement is sufficient or not. However, if you are less that 99% certain, you have a decision to make: should you spend the time to prove that you are right or should you guess and move on. Again, the key here is that many people don’t even realize that they are in fact guessing when they are not close to 100% certain in their answer – they think it is just the nature of DS and that it is fine to be pretty sure on a question and then move on. It IS ok to “go on a hunch” as long as you recognize that you are essentially guessing and that there is a chance, perhaps a good chance, that you are wrong. For the sake of time management, sometimes it is indeed better to guess and move on since in many cases it would take a very long time to prove that a statement is sufficient or not sufficient, but you need to accept and understand that when you pick an answer on DS without a very high level of certainty, you are essentially guessing.

If you have the time and you believe that it is within your ability to go further than the conceptual understanding that you think you have, the next step is to try to actually PROVE that a statement is sufficient or not. Again, I will come back to this in a future post because it will require a fair amount of explanation and some examples to really illustrate how that would work exactly, but for now suffice it to say that in most cases your choices are algebra and/or picking numbers. Questions that are algebraic in nature (either they give you equations or they give you words that could easily be translated into equations) are obviously approachable in an algebraic way, though it must be said that even in these cases sometimes picking numbers is a better approach. And on other questions that have less of an algebraic structure, especially number property questions, are often best approached by picking numbers (and often can’t be done algebraically).

So to summarize, try to force yourself to reach a high level of certainty in your judgment about the statements on a DS question. “Going on a hunch” or being “pretty sure” is just not enough and will often lead to many wrong answers. Sometimes proving things out would be too time consuming or difficult, but at the very least understand that when you are not sure on a Data Sufficiency question you are essentially guessing.

In a future post I will discuss the specifics of “proving” statements to be sufficient or not and will give some examples, so stay tuned!

Understanding the “Spectrum of Sufficiency” is a key advanced strategy on Data Sufficiency questions and yet very few people seem to be aware of the technique. Nevertheless, understanding the concept and how to apply it can really help you answer some of the most difficult DS questions correctly and can also help prevent you from falling into traps.

First, before explaining the concept, let’s begin by discussing an underlying assumption upon which this strategy is based: one of the key features of Data Sufficiency questions is that they are meant to trick test takers and therefore obvious answers on DS questions are almost always wrong (this is true even for most easy and medium level DS questions). Now, it is not always easy to tell what an “obvious” answer is on DS, but as you get more accustomed to DS questions you will start to get a sense for what is just too straightforward and obvious. Furthermore, if you are scoring pretty high on Quant, chances are that the questions you are seeing will be quite difficult, so in that case especially obvious answers will tend to be sucker answers that you don’t want to pick.

Now, enter the “Spectrum of Sufficiency.” Basically, the answer choices on DS questions can be arranged from left to right on a scale of “how much information is needed to answer the question.” In other words, choice D indicates that each statement alone was sufficient, so you really didn’t need that much information to achieve sufficiency. Next to that would be choices A and B, which indicate that one of the statements was sufficient but the other not. Next would be choice C, which of course indicates that you needed yet more information to arrive at sufficiency: both statements were needed together. Finally, you have choice E, which indicates that no matter what you did (using the statements both alone and combined) there was not enough information to answer the question. I personally arrange the answer choices in this order from left to right:

The key here is that if you arrive at what appears to be a somewhat obvious answer, not only can you infer that it is probably wrong, but you can also get some sense for what a probable answer would be based on the “Spectrum.” In other words, if you are considering choice C, but feel like it is a bit too obvious, the right answer is probably just to the right or just to the left of choice C, since its unlikely that you are going to jump completely to the other end of the spectrum (though that definitely does happen in some cases). In fact, I find that people often think that there is not enough information to answer the question when sometimes there is, so with my students I often find myself advising them to “stay to the left.” In other words, when they think the answer is E, it is actually C. Or when they think it is A or B, it is usually D.

Let me give an example to illustrate how powerful this can be. Consider the following question (from the GMATPrep exams):

This is a very difficult question, but if you are skilled at the art of Data Sufficiency then there is a good chance that you can get it right even if you can’t properly “solve” it. What do you think the obvious (read sucker) answer is on this question? The sucker answer is C, because obviously if you have the values of n and k, you can answer the question since then you would know exactly what number is being divided by 10 (some DS neophytes miss even that because they think that they would actually need to solve for the remainder and think that it would be impossible to do that with such a large number, when of course it doesn’t matter what that remainder actually is so long as we know that we would be able to solve for a specific remainder).

Now, if C is not the correct answer (which at this point we can’t say with 100% certainty it is not), what are the next most likely answers. Well, if we look to the right on the Spectrum of Sufficiency, that would put as at choice E, but that is impossible because for sure if we have the values of n and k we could answer the question. So E is out. Looking to the left, we have A or B. It is unlikely that the answer would be D at this point since both statements appear at first to be insufficient (so for them to both all of a sudden become sufficient is unlikely). Therefore, it would be best to consider choices A and B and ask yourself which of the two statements is more likely to be sufficient (given that we know that E is out and that C is possible but extremely unlikely). Well, if you look at the statements, statement 2 really cannot be sufficient because even if we know that it is 3^22 + k, without knowing the value of k we could never know the remainder. Just think about it: 3^22 is some number, but we could add 1 or 2 or 3 or whatever on to that and that will change what the remainder will be when divided by 10 (or any number really).

At this point, you have to look at statement 1 and consider that it is very likely sufficient given that E, B, and D are out (remember that D is out because statement 2 is not sufficient so that eliminates both B and D) and that choice C is just a little too easy and straightforward (it would be too straightforward even if this were an easy DS question, which it is not). So now you can look at choice A not blindly trying to surmise if it is sufficient or not, but with the foreknowledge that it probably IS sufficient – you just need to figure out how. Here is the thing: even if you don’t have the time or the wherewithal to figure it out, I would just guess A and move on. In other words, you could basically guess correctly on this question (and it is a very difficult one at that) without even really knowing how to solve it. That is pretty powerful stuff and it demonstrates how strategy and technique on Data Sufficiency itself are as important or more important than the content that underlies the questions themselves.

I won’t go into a full explanation here of why statement A is sufficient, but just briefly, since n must be a positive integer and since n is multiplied by four, this ensures that the powers will increase by increments of 4 (you could see this by plugging in integer values of n). And there is a pattern in the units digit of powers of 3 such that the units digit is always the same as you advance by jumps of 4 (from, say, 3^6 to 3^10 to 3^14). If you knew exactly how to solve this, that is obviously great, but the truth is that you can answer this question without knowing how to solve it and without even spending the time that it would take to solve it, just by understanding the Spectrum of Sufficiency and the way it relates to the trickery of the question type.

So when you are solving Data Sufficiency questions, be very wary of “obvious” answers and consider the alternatives by using the Spectrum of Sufficiency. Often you will find, as in the above case, that the some of the alternatives can be completely ruled out and this will often put you in a position between two answers: the more straightforward, obvious one and an answer that sits immediately to the left or right on the Spectrum. At that point, go back and consider that that other answer might very well be correct or if you need to guess, avoid the obvious choice and select the most reasonable alternative. Again, based on my experience people much more often think that they don’t have enough information when in fact they actually do, so if you are really stuck and not sure which way to go, take this advice: Stay to the Left!