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Prime Factorization: My single favorite topic on the GMAT. It’s not even a contest.

My passionate (some would say evangelical!) advocacy of prime factorization results not only from my finding prime numbers so inherently fascinating in and of themselves, but also from the plain and simple truth that prime factorization proves surprisingly useful on questions on which prime numbers aren’t even mentioned.

For example, any time you’re given a question asking about multiples and factors, you can bet that prime factorization will help you get to the answer quicker.

Case in point — this Data Sufficiency question from the Official GMAT Guide:

*If positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105?*

*(1) x is a multiple of 9*

*(2) x is a multiple of 25*

Notice, no mention of prime numbers at all. But take any other approach to this problem, and you’re likely to get pretty frustrated and lost rather quickly. You could certainly test numbers, but good luck taking only two minutes finding values that work for every case!

Now, I’m going to re-write the question and statements using only prime factorizations:

*If positive integer x is a multiple of 2*3 and positive integer y is a multiple of 2*7, is xy a multiple of 3*5*7?*

*(1) x is a multiple of 3*3
*

*(2) x is a multiple of 5*5*

All of a sudden, the question becomes much more manageable. We know from the prompt that x carries at least one 2 and one 3 as factors. We also know that y carries at least one 2 and one 7 as factors. Therefore, the product xy must carry at least two 2s, one 3, and one 7. We are asked if xy carries at least one 3, one 5, and one 7 as factors. After reading the prompt, we know xy has one 3 and one 7, so all that’s missing is the one 5.

Notice what we’ve just done: We’ve shown that in order to establish sufficiency, all we need to do is determine whether there’s a factor of 5 somewhere in x or y (or both).

Statement 1 lets us know that x has two 3s and mentions nothing of 5s. But that doesn’t necessarily mean there isn’t a 5 there. There also might be a factor of 5 in y. Because we cannot determine the presence or absence of factors of 5, this statement is insufficient.

Statement 2, on the other hand, lets us know that x definitely has a factor of 5. And again, we already know from the prompt that x has a factor of 3 and y has a factor of 7. Therefore, the product xy has at least one 3, one 5, and one 7 as factors, and we can conclude unequivocally that xy is a multiple of 3*5*7 = 105. Sufficient.

Final answer: B

Even on questions that do explicitly mention prime numbers, things can get really ugly really quickly if you don’t use prime factorization.

For example, take this Problem Solving question, also from the Official Guide (answer choices not included):

*In a certain game, a large container is filled with red, yellow, green, and blue beads worth, respectively, 7, 5, 3, and 2 points each. A number of beads are then removed from the container. If the product of the point values of the removed beads is 147,000, how many red beads were removed?*

The use of 2, 3, 5, and 7 is a prime clue (pun very much intended). You might look at 147,000 and panic because the number is so large. But let’s break down 147,000 into it’s prime factorization:

147,000

= 147 * 1000

= (7 * 21) * 10 * 10 * 10

= (7 * 7 * 3) * (2*5) * (2*5) * (2*5)

Now, the question asks us how many red beads were removed. Red beads are associated with a point value of 7.

We know that the final point total was 147,000, and when we broke that number down, we found that there were only two factors of 7. Therefore, the only way we could get that score is if we removed 2 red beads. That’s it! 2 is our final answer!

These are just two examples of a large number of questions made easier by prime-factor prowess. Practice making those factor trees! And notice how prime numbers help you answer questions about other topics like Greatest Common Factor and Least Common Multiple.

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Data Sufficiency questions are often difficult to get used to, because they require an adjustment in your approach to math problems. When you went through math classes growing up, the end goal was always “Find the value of x” or “Find the area of this circle.” You were asked to give hard responses to these questions, and nothing mattered more than finding a definite value.

With Data Sufficiency, answering the question does not matter as much as *the ability to answer the question*. You are not primarily concerned with the final answer, but rather whether you have enough information to get you to that answer. For example, if you’re asked to find the value of x, and a statement tells you that 300x + 257 = 1345, you know that this statement is sufficient, because you can perform arithmetic on that equation to isolate x. Are you going to perform it? No, because it’s too complicated and you don’t need to! All you’re concerned with is whether you *can* find the answer.

It might strike you as odd, but because of this principle, you can tackle some supposedly difficult DS questions without writing down a single equation or calculation! Sounds too good to be true, but in actuality, it makes a lot of sense. Remember, in business school you’ll be given data in case studies, and you’ll be expected to determine relatively quickly what information is relevant. DS questions are perfect for testing this ability because you have to look at the information given to you and cut to the heart of what is most important about that information.

As an example, let’s look at this rather wordy DS problem:

*Martha bought an armchair and a coffee table at an auction and sold both items at her store. Her gross profit from the purchase and sale of the armchair was what percent greater than her gross profit from the purchase and sale of the coffee table?*

*(1) Martha paid 10 percent more for the armchair than for the coffee table.*

*(2) Martha sold the armchair for 20 percent more than she sold the coffee table.*

First, let’s approach this algebraically to show how cumbersome it ends up being:

In general, Gross Profit (P) is the Selling Price (S) minus the Buying Price (B):

P = S – B

We want to know what percent greater the profit of the armchair (P_armchair) is than the profit of the coffeetable (P_coffeetable). If we represent the missing percent as x, then the equation would be

P_coffeetable*(1 + x/100) = P_armchair

Rearranging, we would get:

x = 100*(P_armchair / P_coffeetable – 1)

We know that P = S – B, so we can substitute:

x = 100*[(S_armchair - B_armchair)/ (S_coffeetable - B_coffeetable) - 1]

Confused yet?? I sure am!!

But try to look at things from a sufficiency point of view. Notice that you need absolute values for the gross profits in order to solve for x.Â You could also find the ratio between the two gross profits.

Now that we’ve seen how ugly this looks when all the algebra is written out, let’s take a more common-sense approach.

The prompt:

*Martha bought an armchair and a coffee table at an auction and sold both items at her store. Her gross profit from the purchase and sale of the armchair was what percent greater than her gross profit from the purchase and sale of the coffee table?*

What we need in order to determine sufficiency:

In essence, this question asks you to compare the values of two profits. You need the value of each profit OR the ratio between the two profits. Notice that you can figure out what information you will need without writing down a single number or algebraic expression.

What each statement tells us:

*(1)Â Martha paid 10 percent more for the armchair than for the coffee table.*

Without writing any math, you can deduce that this is insufficient, because buying prices are mentioned, but no selling prices.Â And with no selling prices, we certainly can’t determine anything about profit.

*(2) Martha sold the armchair for 20 percent more than she sold the coffee table.*

Now, we’ve got information about selling prices, but nothing about buying prices. Again, Insufficient because there is no way to determine profits.

So far, nothing too difficult. It’s pretty simple to narrow this down to C and E.Â But how do we determine whether the statements together are sufficient?

You could test numbers here, but really all you need to do is realize that the statements only give you percentages to work with.Â For the sake of illustration, let’s pick numbers to see what this means:

According to Statement 1, Marta could have spent $100 on the coffee table and $110 on the armchair, or it could have been $10 on the coffee table and $11 on the armchair. (Unlikely prices, maybe, but remember, the real world doesn’t apply here!) There are infinite possibilities for what the buying prices could have been.

Likewise, Statement 2 tells us that Martha could have sold the coffee table for $100 and the armchair for $120. Or it could have been $10 for the coffee table, $12 for the armchair.

You’ll notice that because the absolute numbers for selling and buying prices vary so much, so too do the gross profits! And if the gross profits can fluctuate that drastically, there is no way on Earth you can nail down one specific percentage increase from one profit to the next!

And thus, without writing a single equation, you can determine that the answer must be E.

It’s very very tricky to get your mind to think this way, especially since you’ve been trained all your life to hack away at a problem until you come up with a definite answer. But it is absolutely imperative that you begin to look past the math of DS questions and ask yourself what information is *necessary to solve the problem*.

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In any GMAT prep course, one of the first things taught about the Data Sufficiency section is that *the two statements are true and do not contradict one another*.Â It’s a point that’s easy to gloss over and completely overlook during the hustle and bustle of your test prep.

But this supposedly self-evident point gets many students into trouble when dealing with YES/NO questions, because they mistakenly try to prove or disprove *the statements* rather than *the prompt*.

I’ll explain:Â Recall that a YES/NO question is one in which the answer will be “Yes” or “No.”Â For example, “Is x even?” or “Are the distances equal?”Â This is in contrast to VALUE questions, for which you must come up with one particular value (e.g. “What is x?”, “What is the average of a and b?”).

If a statement produces both a YES and a NO, then it is insufficient.Â If the statement (or combination of statements) always produces a YES or always produces a NO, then it is sufficient.Â (Remember, a NO is not the same thing as INSUFFICIENT; so if you’re asked “Is x even?” and a statement lets you know that x is always odd, then that is SUFFICIENT, because you can answer NO with certainty.)

Basic example:

Is x odd?

(1)Â x is a multiple of 3.

(2)Â x is a multiple of 5.

For Statement (1), x could be 3, which would lead to a YES, but x could also be 6, which would lead to a NO.Â Insufficient.

For Statement (2), x could be 5, which would lead to a YES, but x could also be 10, which would lead to a NO.Â Also insufficient.

Combining the statements, we see that x could be 15, which would lead to a YES, but x could also be 30, which would lead to a NO.Â Â Final answer, E: the statements together are not sufficient to answer the question.

This is a simple example that would not likely appear on the GMAT, but it’s great for illustrating a basic mistake students make: trying to disprove the statements.

It might be tempting to look at Statement (1) and try to find a YES or a NO *to the statement itself*, rather than the prompt.Â So you try to prove/disprove “x is a multiple of 3″, rather than prove/disprove the real question, “Is x odd?”

This would result in you picking, let’s say, x = 3, because it answers YES to “x is a multiple of 3″.Â Then you might pick x = 5, because it answers NO to “x is a multiple of 3.”

But of course, both 3 and 5 answer YES to the question in the prompt, and you may erroneously conclude that Statement (1) is sufficient, when in actuality, it is not.

Obviously, this approach can get you into trouble, because you may get an incorrect answer.Â But there’s an even more basic error behind this mistake:Â You’re wasting valuable time trying to prove/disprove something *that is already known to be true*!

And thus I return to that basic maxim of Data Sufficiency questions:

The statements are *always true* and *never contradict one another*.Â Again, it seems like a trivial point, but as the aforementioned example demonstrates, you’d be surprised how forgetting the basics can lead to unnecessary wasted time!

So, in conclusion, recognize that the statements are true, and use their information to address what really matters:Â the question in the prompt.

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Geometry is an important part of any GMAT test-taker’s conceptual toolkit. On Data Sufficiency geometry questions, it’s especially key to have an intuitive feel for what is and is not solvable given certain bits of information. Consider the following difficult problem:

A circle having center O is inscribed in triangle ABC. What is the measure of angle BAC?

- The radius of the circle is 2.
- Segment OA has length 4.

(A) Statement (1)Â ALONE is sufficient, but statement (2)Â alone is not sufficient to answer the question asked.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

(C) BOTH statements (1) and (2)Â TOGETHER are sufficient to answer the question asked, butÂ NEITHER statement ALONE is sufficient to answer the question asked.

(D) EACH statement ALONE is sufficient to answer the question asked.

(E) Statements (1) and (2)Â TOGETHER areÂ NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

There are two ways to attack a problem like this. At the end of this article is an extremely well-thought out, coldly-reasoned, logical,** academic** explanation. While interesting, in terms of the GMAT it is an **absolutely useless** way to think about the question. It just takes too much time! Instead, you should train yourself to approach problems like these intuitively. Here’s how:

First, check out this simple geometry tool. It’s a handy JAVA applet that lets you *see* how the problem works visually. If you take a few moments to play with the applet, you may be able to get a better *intuitive *feel for the restrictions created by statements (1) and (2).

Try keeping the radius the same while changing the size and shape of the triangle. Notice the angle BAC changing? Next try keeping the length of segment OA the same while moving the circle around to change its radius. See the angle changing again?

By thinking intuitively, you can tell that neither statement is sufficient on its own. When the radius and segment length are fixed, though, it’s another story. Once you know both these pieces of information, you can tell you’re dealing with a 30-60-90 right triangle (more on this below), so finding the measure of angle BAC is a matter working with triangle properties.

That’s why the answer here is C — statements (1) and (2) together are sufficient, but neither is sufficient alone.

The moral here is to avoid wordy reasoning in geometry whenever possible. Practice the art of visualization. You can’t use a nifty applet on test day, but you can draw pictures of extreme cases and move the segments around in your head. This kind of intuitive reasoning is essential on Data Sufficiency geometry questions —where time is short and diagrams are seldom drawn to scale.

Now that you’re thinking visually, take a look at the wordy explanation. Really — do not read the following explanation until you play with the applet! There are tons of sites out there like the one I mentioned above. Spending some time playing with the possible orientations of triangles and circles is going to build your geometric intuition, which will only help your GMAT score.

Here’s the **wordy explanation**. Look out.

Note that at the three points where the circle touches the triangle, the radius of the circle connecting these points to the center of the circle is going to be perpendicular to each of the triangle’s sides.

Statement 1 tells us that the radius of the circle is 2. Although this defines the circle entirely, there are many possible triangles in which a circle of radius 2 could be inscribed — imagine that the triangle that this circle sits inside of is NOT an equilateral triangle. It is certainly possible that triangle ABC is scalene, in which case each angle has a different measure. Since Statement 1 makes no mention of any of the triangle’s vertices, angles, or sides, note that by simply relabeling the vertices of a scalene version of triangle ABC, we could have different measures of angle BAC. Statement 1 is not sufficient.

Statement 2 tells us that segment OA has length 4. Although this gives us more specific information about the vertex named in angle BAC, note that, again, this statement makes no restrictions on the measure of the angle BAC. Imagine a really skinny acute triangle where angle BAC is very small, and the distance between point A and the segment BC is very close to the length of segment OA = 4. Or imagine a really fat obtuse triangle where angle BAC is very large, and the distance between point A and the segment BC is just slightly less than twice the length of segment OA = 4. In the first case, the measure of angle BAC is small, in the second case, the measure of angle BAC is close to 180 degrees. Statement 2 is not sufficient.

Taken together, the two statements tell us two important pieces of information about the smaller triangle AOD, where D is the point where the circle touches segment AC. Segment OD is a radius of the circle, so from statement 1 we know that it has length 2, and statement 2 tells us that OA has length 4. Knowing that the angle ODA is 90 degrees (remember that the radii of the circle are perpendicular to the triangle at each of the points of tangency) tells us that the triangle OAD is a right triangle. Furthermore, we know that the hypotenuse of the right triangle, OA, is twice the length of the leg, OD, the radius of the circle of length 2. This is beginning to look a lot like a 30-60-90 triangle. Noting that it would violate the triangle inequality if the leg DA were less than or equal to the length of leg OD, we can conclude that the leg OD = the radius of the circle is the shortest leg of the triangle, and we do indeed have a 30-60-90 triangle.

Now we can conclude that angle OAD is equal to 30 degrees, being opposite the shortest leg, but what about angle OAE? We need to know the measure of this angle in order to find the measure of angle BAC. Symmetry applies here. All statements applied to segments OD and DA also apply to OE and EA because triangles AOD and AOE are congruent — they share side OA, angles OEA and ODA are equal (right angles), and the triangles have their shortest sides equal to the radius of the circle. Hence the measure of angle BAC is twice the measure of angle OAD: 2 Ã— 30 = 60. The statements together are sufficient.

]]>The GMAT quantitative section is different from most math tests. You don’t usually see Data Sufficiency questions outside the GMAT, for one thing. They’re tricky, and mastering them requires a high level of familiarity. The good news is that the answer choices are the same for every question, and precise calculations are often unnecessary.

Then there are the word problems. All that text takes a long time to read. With 37 questions to do within a scant 75-minute period, you have an average of about two minutes to answer each question. It can be nerve-racking to spend almost half of this precious time just parsing out questions that are essentially prose versions of a company’s balance sheet.

*photo by stuartpilbrow*

Maybe it seems silly to you to have to read through a lengthy explanation of two trains traveling on parallel tracks at different rates, when it would be a lot simpler to just look at a well-labeled diagram. After all, there is a reason why balance sheets, graphs, and diagrams exist, right?

There is a reason behind the test-maker’s strategy, however. These questions are testing how well you can take information that’s disorganized, messy, and portrayed in a slightly illogical way, and turn it into a *correct decision.* As a banker, manager, CEO, COO, CFO or any other leader in the world of business, you’ll have to make decisions based on information from people that “report” to you. It’s likely that these people won’t have the type of intense training you’ll have had at business school. Chances are, they’ll communicate with you in a somewhat disorganized, messy, and slightly illogical way.

How do you answer more GMAT quantitative questions correctly? Take the same steps that a good businessperson would take in order to make a decision:

- Calmly and carefully obtain information.
- Think analytically.
- Decide without second-guessing yourself.

Though all three steps are required for all GMAT math questions, the bulk of the work required for overly-wordy math problems comes in the first step: calmly and carefully obtaining information. It doesn’t pay to read the prompt and answer choices as quickly as possible. Getting the question correct requires a thorough knowledge of all information in the prompt. Skimming will only increase your risk of misreading or omitting important facts. Consider the following question, taken from the 2nd edition of the GMAT “Quantitative Review”:

One week a certain truck rental lot had a total of 20 trucks, all of which were on the lot Monday morning. If 50 percent of the trucks that were rented out during the week were returned to the lot on or before Saturday morning of that week, and if there were at least 12 trucks on the lot that Saturday morning, what is the greatest number of different trucks that could have been rented out during the week?

A: 18

B: 16

C: 12

D: 8

E: 4

This long-winded question boils down to:

Find the maximum value for t if (1/2)t + (20 – t) â‰¥ 12

This inequality is easily reduced using algebra (pop quiz: solve this.) The tasks of reading the question and interpreting it into a suitable equation are a lot more time-consuming. If you felt rushed and skipped the simple phrase “of that week,” then the incorrect answer choices C and D would become far more compelling than if you had read the question accurately.

The key here is to read the question as if you had all the time in the world—*the first time through.* Read it carefully. Absorb every word. Write down expressions and equations. Misreading a question will either lead to an incorrect response or a reread. Having to reread a question means that your first reading was a waste of time, and wasted time, much like an incorrect response, will always add up to a lower score.