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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*.