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 Invisible Gorillas: Prime Factors

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In the last few blog posts we’ve explored “Invisible Gorillas”. As mentioned in the original post, Invisible Gorillas are patterns that recur on standardized tests that you can easily miss if you’re not looking for them. However, they’re simple to spot if you’re on the lookout for where they might show themselves.

 

I am not old. Granted, I didn’t get my first cell phone until I was in college. And yes, the first music I ever bought was on cassette tape. And sure, Google didn’t exist when I was born, but still. I am not being old. So why didn’t my teachers cover prime factorization when I was in school?

 

It’s not like the concept of prime factors is anything new. I would give you some historical evidence of this, but this newfangled Google thing gives me some trouble sometimes. But still, why wasn’t prime factorization more prevalent in schools when I was growing up?

 

For those of you as old as I am (or at least as disadvantaged as I was in not getting this information in school) prime factorization says that we can break any number down to smaller pieces—it’s prime bits—and make it easier to handle. We do that by first taking any factor pair and then seeing whether the factors are prime, or whether they are non-prime, in which case we repeat the process until we have all primes. Take the following example:

 

 

Our analysis shows us that 84=2*2*3*7. This in itself seems pretty simple, but making things simple is often the major step that you need to take in order to solve what looks like a complex problem. Take the following for example:

 

 is a factor of 540. What is the greatest possible value of x+y?

 

Although the form makes this look daunting, it’s simply an exercise in prime factoring. Watch what happens when we re-write the information in that form:

 

 

So, 540= 2*2*3*3*3*5. All that’s left to do is re-write that in a form that more closely resembles the form in which the problem was given.

 

 

Once the number has been broken into its prime factors, it’s plain to see that we can only squeeze two twos and three threes out of this number. There’s no different way we could break it down to get more twos or threes. Prime factoring figures out how many pieces of each type we have and allows us to re-assemble them in any way we see fit. So the greatest possible x is 2 and the greatest possible y is 3. Thus, 2+3 is equal to our answer of five.

 

Whether you’re old or young or somewhere in between, jump on board with prime factorization because taking numbers and breaking them into their component prime parts is a great way to solve problems and a recurring pattern on your test that you’ll be sure to recognize if you look for it!

 Invisible Gorillas: Answer Choices in Different Forms

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In the last few blog posts we’ve explored “Invisible Gorillas”. As mentioned in the original post, Invisible Gorillas are patterns that recur on standardized tests that you can easily miss if you’re not looking for them. However, they’re simple to spot if you’re on the lookout for where they might show themselves.

 

Today we’re looking at the form of the answer choices. It’s a frustrating experience to feel that you’ve done every step correctly in a math problem and to reach the end only to realize that your choice isn’t there. Self-doubt can quickly creep in leading to thoughts of “Where did I mess up?”, “Do I need to start over?” or “Whyyyyy is this test sooo harrrrrrrd?”

 

Avoid the whininess of that last question and realize that having the value you reached at the end of your work not line up with the values in the answer choices is a common experience, and it doesn’t mean that you did anything wrong. Often by slightly transforming the answer choices the testmaker can take a simple question and make it one capable of separating the unprepared student and the test-taking superstar (that’s about to be you!).

 

Here are three common transformations that you’re likely to see.

 

  1. The answer is a combination of variables- Perhaps there’s a fact pattern that’s set up to have you solve for x. After going through all the math you’re confident that you’ve correctly solved for x, but you don’t see the value that you got. Double check to make sure that the question doesn’t ask for some form of x, such as 2x or x+y. It’s hard to get the right answer when you’re answering the wrong question!

 

  1. The answer is a reduced fraction- This is perhaps the simplest one on the list, but the correct answer to the problem will be the fraction reduced to it’s simplest form. If your answer is 18/51 and you don’t see that in the choices, don’t worry. The testmaker’s version—6/17—is likely just a reduced form of the fraction that you didn’t initially see.

 

  1. Your answer has a radical at the bottom of a fraction- For some reason many students fail to see that the following is true:

 

 

If you’re left with a square root in the denominator of a fraction, simply multiply both the top and bottom of your fraction by the same square root in order to simplify. Remember, that by the definition of a square root when you have this:

 

 

You can re-write it as:

 

And cancelling will leave you with:

For example:

 

 

Keep these examples in mind as you practice and you’ll realize that some of those times when you start to worry that you’ve gotten the wrong answer are just situations where you haven’t recognized your correct answer in another form!

 The Day Before the Test

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In honor of several of my tutoring students who are taking the GMAT in the next two weeks, I’m taking a break from the Invisible Gorillas series in order to cover some practical advice for what to do the day before your test. In order to do that, I want to tell you a story about what happened the morning that I took my LSAT.

 

I was attending school at UC San Diego and living on campus (because I was a resident advisor) in the fall of 2005 when I took my LSAT. Living on campus meant being able to stagger out of bed and into class in mere minutes, so I was a little nervous about making it all the way across town to the University of San Diego’s law school in order to take my test. My test information said I was supposed to arrive between 8 and 8:30. I planned on leaving at 7:30 to make the 15-minute drive comfortably.

 

I woke up even before my alarm went off, an occurrence virtually unknown to my college self. I took a shower, got dressed, had my breakfast and got my things together well ahead of schedule. I was confident about the test, but my nerves were going with the natural anticipation of the big day. Having double checked that I had all the proper documents I walked out to my car to make the drive.

 

I sat down, checked that I had everything once more and turned the key.

 

Silence.

 

I made sure the car was in park, that my foot was on the brake and that the key was pressed in completely.

 

I turned the key again.

 

Silence.

 

I don’t know if you’ve ever had a “this can’t be happening to me moment” but if you have you know that the surrealism of the moment makes practical action almost impossible. My completely reliable car that had never failed to start in the three years that I’d owned it had chosen this day to break its streak because of an alternator with impossibly poor timing.

 

If you’ve ever lived on a college campus you’re probably aware that 7:30 a.m. on a Saturday is not the most active time. Almost everyone is asleep or gone for the weekend so the possibilities for catching a ride are not numerous. Luckily, I called the one friend I knew who was on campus, had a car, and was generous enough that he just might let me borrow it for the morning. He completely bailed me out and I was able to make it to my test (Thanks again Niheer!).

 

The moral of the story is that test day can be stressful. Sometimes it can be very stressful before the test even begins. For that reason, the day before the test is time to relax. Working one more problem set or reading one more passage probably isn’t going to make the difference between a great score and a mediocre one. It very well might stress you out and make you worry unnecessarily.

 

The day before the test is not a day to study. It’s a day to remind yourself of the progress you’ve made from the start of your journey until this point. It’s a day to visualize test day going well. It’s a day to reduce your stress levels so that you’re ready to take on whatever the next day will bring.

 

So, the day before your test go watch a movie. Have dinner with friends. Take a nice relaxing walk. And maybe, just maybe, you should make sure your alternator is working properly. But whatever you do, relax.

 Invisible Gorillas: Too Much Information

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In the last few blog posts we’ve explored “Invisible Gorillas”. As mentioned in the original post, Invisible Gorillas are patterns that recur on standardized tests that you can easily miss if you’re not looking for them. However, they’re simple to spot if you’re on the lookout for where they might show themselves.

 

While the last few Invisible Gorillas that we’ve covered have been mathematical patterns that you’re likely to encounter, this one is going to be a little more general: The Impossible Numbers Situation. Believe it or not, the goal of the testmaker is not to completely demoralize you (despite how it may seem). With that in mind, know that all the problems you are being given are meant to be solved. So, panic is NOT the correct response when you see the following:

 

The integer N is equal to the cube root of 187,311 rounded to the nearest integer plus 18! multiplied by the product of all prime numbers less than 25. Which of the following must be true?

 

  1. I.                   N is even
  2. II.                 N is prime
  3. III.               N is a multiple of 3

 

Remember, panic is NOT the approach we’re hoping for. Consider this paradoxical truth about standardized testing: If it seems impossible, there’s probably a simple approach.

 

The simple approach here is quite clearly NOT to calculate N and use that to evaluate the three statements. For purposes of illustrating that, if you’d bothered to calculate you would have found that N=1,428,323,924,823,407,675,653,590. Even if your test allows you to use a calculator, I hope we can agree that is a number that no one should have to calculate.

 

But, rather than go to “there’s no way I can calculate that, it’s hopeless” you should jump to “there’s no way I can calculate that… there must be an easier way!”. Evaluating this problem with that thought process in mind yields the results that we want.

 

  1. I.                                N is even

 

Okay, so I can’t find N, but how else can I find whether N is even? I can look to see if N is the product of some number and 2. And in fact, since we’re multiplying by the product of all prime numbers less than 25, and 2 is one of those numbers. We know that N can be re-written as “some really big number times two”, so N must be even!

 

  1. II.                             N is prime

 

Again, all I know is that N is a really big number, but I do know something about what primes are. A prime is simply a number that has no factors other than one and itself. Since N is defined as the product of two integers, it must not be prime.

 

  1. III.                           N is a multiple of 3

 

By using the same logic as we did for the first statement, we can see that since 3 is a prime number less than 25, and since we’re multiplying by those primes as part of this process, 3 must be a factor.

 

As you work through your practice problems, keep an eye out for the impossible problem. The more impossible it looks, the more likely it is that there’s some simple way to solve it just sitting there waiting for you.