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.
]]>Do you bite your nails? Chew on pencils? Forget to the check the subject and verb on sentence correction questions? All of these are bad habits, but only one will affect your GMAT score. Test-takers tend to make the same grammar mistakes over and over again; learn to recognize—and avoid—these common traps and pitfalls.
1) Ignoring the Subject and Verb
It’s one of the biggest, most basic rules of grammar: If it doesn’t have a main subject and main verb, it’s not a sentence. More importantly, the subject and verb are the potential home of numerous SC errors, including subject-verb agreement, sentence fragments, clause connection and more. Many test-takers head straight for more complicated issues and overlook an easy subject-verb mistake that can knock a few answer choices out of contention. In a gourmet meal, perfect side dishes don’t matter if the main course is burned. Think of the subject and verb as the meat (or vegan substitute) of a sentence, and check them first.
2) Overlooking Redundancy
The GMAT loves concision and clarity; it is a mistake to repeat yourself and say the same thing twice. Did you see the redundancy in the previous sentence? If so, you are ahead of the curve! The vast majority of test-takers miss simple redundancy errors, so be extra vigilant in watching out for them. Look for quantity words:
Redundant: The price dropped by a 30% decrease.
Awesomely concise: The price dropped by 30%.
Redundancy can also appear in cause and effect relationships:
Redundant: Because she is obsessed with Elvis, the result is that she eats only peanut butter and banana sandwiches.
Awesomely concise: Because she is obsessed with Elvis, she eats only peanut butter and banana sandwiches.
The moral of the story: Saying it twice? Not so nice.
3) Trusting Your Ear Too Much (or Not Enough)
Listen up, native English speakers: Your natural ear for grammar can be a hugely valuable tool. Use it, and trust it to help you eliminate obviously awkward answer choices. Don’t waste time analyzing sentences that you instinctively know sound wrong. However, trusting your ear is not a substitute for understanding the rules of grammar. This is especially true for GMAT idioms. Take a look at this sentence:
As puppies gnaw on their favorite bones, so I gnaw on this ear of corn.
This sentence might sound totally fine — in fact, it’s likely that many native English speakers would say this in everyday speech, especially when enjoying a delicious corn-on-the-cob. Unfortunately, “as…so” is NOT a valid comparison idiom. The correct sentence is:
Just as puppies gnaw on their favorite bones, so I gnaw on this ear of corn.
The GMAT test-makers understand that a natural ear for English is an advantage, so they will purposefully include sentences that sound wrong, but are actually correct.
4) Playing the “What If” Game
The “what if” game is one of the most dangerous habits on the SC section. You read an answer choice and think, “I see an error here. This is wrong. But what if it had a plural verb (or a comma, or a pronoun, or some other grammatical element)? Then would it be correct?” Do not fall into this trap! It doesn’t matter whether the answer choice would be correct if written differently; it’s not, and that’s the whole point. Answer choices on the GMAT are constructed deliberately and precisely, so don’t waste time with “what ifs.” If you see an error, simply eliminate the choice and move on.
5) Forgetting the Logic of a Sentence
There are so many grammar rules to remember that test-takers often forget that sentences have to make sense, too. If you’re stuck on a question, take a step back and do a quick logic check: What information is this sentence trying to convey, and what is the clearest, most straightforward way in which to convey it? Logic checks are particularly useful for modifier and verb tense errors. For modifiers, ask yourself: What is this phrase describing, and does the structure of the sentence make that clear? For verb tense, ask: In what order did these events occur, and do the tenses correctly express that relationship? Thinking about the meaning of a sentence can help pull the underlying grammar issues into sharper focus.
Ditch these bad habits and say hello to a higher verbal score. As for biting your nails… you’re on your own.
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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. 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: Recall that a YES/NO question is one in which the answer will be “Yes” or “No.” For example, “Is x even?” or “Are the distances equal?” 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. If the statement (or combination of statements) always produces a YES or always produces a NO, then it is sufficient. (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) x is a multiple of 3.
(2) 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. 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. 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. 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. 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″. 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. But there’s an even more basic error behind this mistake: 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. 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: the question in the prompt.
]]>We’ve received grades all our lives. In fact, we’re so used to them that we often don’t think very much about what they mean, or how they are calculated. So today we’re going to look at some of the different ways in which tests are scored, and at what those scores mean.
In preschool, we receive grades in the form of category scores: gold stars, silver stars, or bronze stars. Sometimes we might get two gold stars, or even three gold stars. These kinds of grades divide the relevant universe of people into some small number of categories, usually low-medium-high.
Later on we start to receive simple tally scores: 8/10 or 23/25. Soon these are represented as percentages: 80% correct, or 92%. One of the funny things about grades is that by the time we’re in high school and college, grades have reverted back to category scores (A, B, C, D, F) through a transformation of the percentages.
Every teacher and school adopts slightly different transformations. In some places, a grade of A is reserved for 96% and above. In other places the cutoff is 92%. In still others, it might be 90%. So what an “A” means can vary widely from place to place.
Everyone knows that some test questions are more difficult than others. Occasionally, teachers will take this into account by awarding more points for the hard questions than for the easy ones.
The basic sequence for most kinds of scoring is this:
For those of you taking the GMAT, the basic sequence is very different. Because the GMAT is an adaptive test, it looks at your performance on each question as you respond to it, and estimates your math or verbal ability along the way. Then it uses that ability estimate to calculate your score. For the GMAT, the basic sequence is:
What the GMAT does explicitly is what all tests try to do implicitly, namely, try to ascertain what you know and are able to do, in some context or another. It’s a more responsive way of testing, and we use the same adaptive technology in our GMAT practice tests.
In a later post, we’ll talk about validity, which has to do with what your score really means within a context, and why anyone would care.
Until then, do your homework!
]]>You already know that regular exercise is good for your physical health. It’s good for your mental health, too. The sense of energetic well-being you get from being in shape extends to your mind, not just your body. Staying fit helps keep you more alert, focused and positive. The exercise itself is a good rest for your mind. Activities like running or brisk walking require enough of your brain to keep it engaged, but don’t ask it to do much heavy lifting. Your brain finds this state very congenial.
If you don’t have an exercise routine, you can still learn some simple, easy techniques to relax and stretch your joints and muscles. You’ll find that any neck-down relaxation technique will also work wonders from the neck up. One easy method is to stand on one foot. Focus your gaze at some distant point directly in front of you. Press your palms together at around chest height. Place your free foot alongside the opposite leg. Try to breathe deeply and focus on your balance. The idea isn’t to challenge yourself acrobatically; if you’re windmilling your arms crazily to stay balanced, it defeats the purpose. If you’re having trouble staying balanced on one foot, keep the other one close to or lightly touching the floor. When the leg you’re standing on gets tired, switch to the other one.
Another easy relaxation technique is to lie stretched fully out on the floor. Use a pillow for your head, but otherwise, the flatter the surface you’re lying on, the better. Focus your attention on each muscle and joint, starting at your toes. Consciously and actively make yourself relax them. Breathe slowly all the way in and all the way out. You can also lie on your stomach with your head facing right or left, it stretches your neck muscles out naturally. Take ten deep breaths and switch directions.
For many more relaxation techniques, check out “Conquering Test Anxiety” by Dr Neil Fiore and Susan Pescar. Get in the habit of practicing relaxation techniques early. Make them a part of the routine. Spend fifteen minutes doing them each day, and they’ll become effortlessly automatic.
Sleep is usually low on the priority list for students. But it’s worth making the time to keep yourself well-rested. After a certain point, sacrificing sleep for more study hours is counter-productive. It won’t matter how prepared you are on test day if you’re too groggy to pay attention. Sleep maximizes the benefit of your study time, because your brain uses the time to organize and process new information you’ve taken on that day – this is one theory about the purpose of REM sleep and dreams.
It’s also important to eat right. Junk food is convenient for late-night study sessions, but it won’t serve you well when it’s time to harness your brain power. Especially on test day itself, go easy on the caffeine and sugar. Caffeine is a natural antidepressant, but too much of it can make it difficult to pay attention. The energy in fruits and veggies metabolizes more slowly and naturally, and its good effects last longer. A happy stomach makes for a happy mind.
Final prescription: Study hard, and take care of yourself well!
]]>Nate Burke is a Content Developer at Knewton, specializing in GMAT prep.
Are you stubborn? There are many situations in which stubbornness would help a human being. A stubborn nature can be extremely useful when trying to accomplish a complicated goal over an extended period of time. Building a house, traveling a long distance, hunting for food, and courting someone are all activities central to human history that require, at least to some degree, a knock-down, drag-out, brick-headed resolve to get the damn thing done NO MATTER WHAT.
It is natural, therefore, that you would want to spend 5 minutes on the first question in the GMAT quant section. It’s okay. Natural and cultural forces have optimized our problem-solving heuristics in a certain way; recognizing which ones actually are optimal in certain situations is the key to good performance. In any situation in life OTHER than the GMAT quant section, thinking really hard and creatively about a problem until a solution is found (even if it is for an extended period of time) will usually be of value. Not on the GMAT.
We have a saying here at Knewton–answer when you are 90% sure, and move on. This is easy to say and extraordinarily difficult to do–precisely for the reasons outlined above. The reason why it is absolutely necessary, though, has to do with optimization. When you take a computer-adaptive test, you are given a powerful tool–the test itself. The test can work in your favor if you allow it to. The idea is that, over time, the test will be able to calculate your ability and give you questions that reflect it. If you find that you are given a particularly tricky question, DO NOT TAKE IT PERSONALLY. Do not assume that you are stupid for not having any idea how to approach it. Do not interpret it as a challenge that must be overcome by sheer intellectual willpower. The question was generated by the computer as the next step in its program to determine where you fall on the curve. The computer generated the question with the expectation that you’d spend roughly 2 minutes on it and move onto the next question. If your approach creates a situation that in any way deviates from these basic assumptions, the algorithm is designed to output a score that is a poor approximation of your abilities–and it will always err on the side of portraying you as stupid.
The long and short of it then is to focus your stubborn impulses in useful ways. Be stubborn about maintaining a rigorous study schedule. Be stubborn about attending your live Knewton GMAT classes. Be stubborn about learning and memorizing exponent rules and common powers. Be stubborn about studying the both the explanations of practice questions you answered incorrectly AS WELL AS practice questions that, while answered correctly, still gave you a tough time. Be stubborn about timing yourself on the easiest questions and work to improve the time it takes to answer them. Be stubborn about spending three weeks waking up at the time you will be waking up on test day to avoid being overly tired. Be stubborn about taking the test again if you happen to have a bad day. In short, be stubborn about improving your score on every single day leading up to the test.
But on test day, let it go. Just answer the question and move on.
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By Jen Rugani
Certain words are very commonly confused for each other; this is by design. Here’s a core content piece you’d find in our course.
The preposition among takes an object made up of more than two items, while the preposition between takes an object make up of exactly two items.
As and like can both be used to suggest similarity between two words or word groups. However, certain situations require the use of like, and others require the use of as.
Like draws a comparison between two nouns and functions as a preposition meaning “similar to.”
As indicates a similarity between clauses, two phrases containing both a subject and a verb. As also introduces an adjective that is used in a comparison or a noun that indicates status.
The adjective fewer properly modifies nouns that express quantities that can be counted; lessmodifies nouns that express quantities that cannot be counted.
Effect is most commonly used as a noun meaning “result,” but may also function as a verb meaning “bring about.” Affect is most commonly used as a verb meaning “change.”
There are several other pairs and sets of words that are commonly confused. The following list is not exhaustive, but it is a good starting point.
Say a friend tells you a story about something that happened to her at work. Her boss has decided to assign her to a very specific research assignment, despite the fact that such research rarely yields practical results. Hearing this reminds you of something that recently happened to you: Your theater director cast you in a performance that was performed even though it was unlikely to draw an audience.
Are these situations identical? Nope! But they are analogous, because they share a few key structural and logical elements. On the GMAT, both Reading Comprehension and Critical Reasoning questions might ask you to identify a situation that is analogous to one presented in a prompt or passage. This requires you to look past the specific details of a situation and focus instead on its underlying logical aspects – things like changes, conflicts, cause-effect relationships, and patterns of reasoning.
Let’s take a look at an example:
Passage Situation: Ectothermic animals—those that cannot control their body temperatures except through external means—often have developed behavioral adaptations that seem counterintuitive. The Peninsular Rock Agama, an Indian lizard, must sit completely exposed on heated rocks for extended periods each morning to reverse the cooling experienced the previous night, flattening its body to the rock when predatory birds fly overhead.
Abstract Situation: An animal cannot maintain its body temperature internally, so it does so environmentally, even though this puts it in danger.
Underlying Aspects: A reliance on the outside environment is necessary, but dangerous.
Analogous Situation: A shopkeeper must publicly post his store’s prices in order to draw customers, although this enables larger stores to easily undercut those prices.
Notice that analogous does not mean identical – a shopkeeper doesn’t seem to have anything to do with a lizard. This makes analogous situation questions particularly difficult for test-takers, because at first glance, the correct answer might seem out of scope.
Don’t let the specific context of the situation throw you; focus on the structural and logical aspects of the passage or argument and find the answer choice that keeps those consistent. You’ll be an analogous situations expert in no time!
]]>Let’s take a look at the following Official Guide DS problem:
If p and q are positive integers and pq = 24, what is the value of p?
(1) q/6 is an integer
(2) p/2 is an integer
It just so happens that this is a value-based question: When we’re asked “what is the value of p?”, we’re being asked to provide a single value for p. If a statement or combination of statements cannot give us a single value, then it must be insufficient.
So, before you go all crazy trying to set up equations, think about what I like to call “The Speedy Road to Insufficiency.” What does that mean? Well, in a nutshell, it means that when approaching a statement or combination of statements, you should treat it as “insufficient until proven sufficient.” In other words, go into it trying to demonstrate insufficiency. Why? Well, in short, because it’s faster. How so? Well, in order to prove insufficiency, all we have to do is find two different possible values for p that satisfy the statement(s). If we can quickly locate two such values, we don’t have to do any more work. We know the statement must be insufficient.
Can we do this for Statement (1)? Well, if q/6 is an integer and pq = 24, then q could be 6, 12, or 24, and the corresponding values of p would be 4, 2, and 1, respectively. We just found three possible values of p. Guess what? We’re done with Statement (1). Definitely insufficient. And in truth, you could have stopped the moment you realized p could be 4 or 2. Two possible values of p are enough to demonstrate insufficiency.
What about Statement (2)? Well, if p/2 is an integer and pq = 24, then p could be 2, 4, 6, 8, 12, or 24. Definitely not just one value of p, so we know Statement (2) is insufficient.
This strategy also works for “Yes/No” questions, which include phrases such as “Is x odd?” or “Is y > 2?” These questions don’t ask for a specific value but instead ask you to answer “yes” or “no” to a specific question. But the strategy is the same: “Insufficient until proven sufficient.” If you can show quickly that the answer could be either “yes” or “no”, you’ve shown insufficiency and can move on. See if you can apply the strategy to this official problem:
If x≠ -y, is (x-y) / (x+y) > 1 ?
(1) x > 0
(2) y < 0
Don’t just post your solution! Also include the way you made things speedy.
This post was written by Rich Zwelling, one of Knewton’s GMAT prep experts.
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