Disguising – and Decoding – Quant Problems
How can the GMAT disguise a prime number (or any other) problem? That’s exactly what we’re going to discuss today! We’re going to use the concept of prime to describe this, but the general process of disguising “ and studying how to decode “ problems is applicable to a great number of problems on the test. Pay particular attention to the end of the article; you can use these concepts when studying a number of different GMAT content areas.
What is a prime number?
First, in order to recognize a disguise, you actually have to understand the fundamental concept at play. So let’s talk about prime numbers.
The concept of prime, by definition, applies only to positive numbers. Also, by definition, all primes are integers. We’re concerned, then, with the set of positive integers (and that’s the only set we’ll consider for the rest of the article).
Most people, when asked to define or describe the concept of prime, will say that a prime number is divisible by itself and 1. (What does divisible mean? One number is divisible by another when the result of that division is an integer. For example, 12 is divisible by 6 because 12 divided by 6 equals an integer.)
Interestingly, though, any positive integer is divisible by itself and 1. The number 1 is divisible by itself and 1. The number 2 is divisible by itself and 1. The number 9 is divisible by itself and 1. Yet only one of those numbers is actually a prime number. If ALL positive integers are divisible by themselves and one, then what is different about prime numbers?
Articulate your answer to yourself before you keep reading.
When considering positive integers, there are three categories: prime, non-prime (or composite), and the number 1. Although the number 1 is divisible by itself and 1, itself equals one. The number 1, then, has only one factor. A prime number is divisible by itself and 1, where itself is a different number than the number 1. In other words, a prime number has exactly two factors. A composite number is divisible by itself, 1, and at least one other number; a composite number, then, has more than two factors (itself, 1, and at least one number in between).
So a prime number has exactly two factors.
(Hmm. What’s a factor? A factor is a positive integer that divides evenly into an integer. For example, 6 is a factor of 12 because 12/6 = 2, and 2 is an integer. In fact, 1, 2, 3, 4, 6, and 12 are all factors of 12 because, when 12 is divided by each of those factors, the result is an integer.)
Okay, so a prime number has exactly two factors. Non-prime numbers have either one factor (the number 1) or more than two factors (composite numbers).
The number 1 is the only number that has exactly one factor. It is not prime. The numbers 2, 3, and 5 all have exactly two factors; they are all prime numbers. The numbers 4, 9, and 16 all have more than two factors; they are all composite numbers.
Okay, so a prime number has exactly two factors, itself and one, while a composite number has more than two factors: itself, one, and at least one other factor between itself and one. That’s the real distinction between prime and composite numbers (and these two categories include all positive integers with the exception of the number 1).
How will the GMAT disguise a prime number problem?
There are a number of ways that the GMAT can test our knowledge of some concept in general. One common theme on more difficult problems, though, is an attempt to disguise the fact that the problem is about (in this case) prime numbers. Often, the word prime will not even appear in the problem. Essentially, the test writers are testing whether you can decode the language to realize that the problem is really asking about prime (or composite) numbers.
For example, a data sufficiency problem might tell us about the positive integer x, which does not equal 1. Are there two integers, both of which are greater than 1, that can multiply to give us x?
What am I really asking you there? Can you think of a specific value for x that would allow you to answer yes to that question? And a specific value that would give a no answer instead?
Let’s see. 2*2 = 4. Those are two integers (they’re not different but notice that my question didn’t specify two distinct integers) and both are greater than 1. They multiple to equal 4. So, if positive integer x is 4, then yes: there are two numbers, both greater than 1, that multiply to give us 4. (These two integers that we’re multiplying are called factors of x. 2 is a factor of 4.)
On the other hand, 1*5 = 5. That doesn’t work because both numbers are not greater than 1, so the answer to the question (for this specific example) is no. For the integer 5, the answer will always be no. Why? Because there’s no way to express 5 as the product of two integers without using 1; there are no factors of 5 that are between 1 and 5.
What’s the difference between those two numbers? The first is composite and the second is prime. A composite number will always result in a yes answer for this question, because a composite number, by definition, has at least one factor between 1 and itself. A prime number, by contrast, will always result in a no answer for this question, because a prime number has exactly two factors, 1 and itself.
Back to my question: Are there two integers, both of which are greater than 1, that can multiply to give us x? I’m really asking you whether x is prime or composite (or, because this is data sufficiency, whether you can’t tell at all). In formulating my question, I incorporated the distinction between prime and composite numbers: I used the definition so that I could avoid giving you the term prime in the question.
Why is this broadly applicable to the GMAT?
This is one of the fundamental ways in which the test writers can make any question harder: ask you about a concept without using the actual name for that concept. It’s not enough just to know the concept; you have to figure out that they’re referring to that concept in the first place.
Your task as a student is to figure out how the test writers can ask about various properties or principles without using the names of those properties or principles. The only way to do this is to give the definition instead, so study HOW that will look on a GMAT Problem. What is the definition, as precisely as possible? What are the different ways in which that could be worded in a question? Can you find two or three OG (Official Guide) questions that ask about the same fundamental principle using somewhat different wording?
Check out #39 from the Quant Review 2nd Edition and #153 from the Official Guide 11th Edition. (Do they look familiar in some way?) The structure of the two problems is almost identical, though the harder one also incorporates additional concepts. In addition, you could solve the easier one by testing real numbers, so you don’t absolutely have to figure out the “prime” aspect in order to answer that one. For the harder one, though, you do actually have to recognize that the problem is testing the concept of prime. Here’s the kicker: it’s easier to figure out the “prime disguise” on the easier problem; then, all you have to do is recognize it on the harder problem. That ability to recognize the fundamental issue will then give you time to deal with all of the additional complexity in the harder problem.
Take-Aways
1) Know the definition of prime, but also know the difference between prime and non-prime (or composite) numbers.
2) Know how the test writers can ask about prime without using that specific word. Don’t stop with prime! What other concepts can they ask about without using the specific word? What are the very precise definitions, and what are the different ways in which they might word a problem to reflect each precise definition?
3) How will you recognize similar wording on future problems?