Decoding Divisibility and Primes on the GMAT – Part 1
Most of my students are driven crazy by GMAT Number Properties. On the face of it, the topic seems straightforward: I know what positive and negative, odd and even are. Divisibility stuff is a little more complicated, but come on: this was taught in school when we were 10! How hard can it be?
Plenty hard, it turns out. The GMAT obviously can’t test you on what you were taught when you were 10; that’d be way too easy. So they have to find some way to make things conceptually harder—and they have definitely succeeded on Number Properties. (I secretly admire how good they are at testing NP, actually. I just don’t like to admit it.)
So we’re going to dive into a series of NP problems to see how they mess with us. We’ll focus specifically on divisibility and prime, the topic that tends to be the most tricky. Try this GMATPrep® problem from the free exams and then we’ll talk!
“*If x, y, and z are integers greater than 1, what is the value of x + y + z?
“(1) xyz = 70
“(2)
”
”Got your answer? Okay, first, let’s understand what’s going on.
Glance: DS. Three variables. The question is a combo (that is, I don’t necessarily have to find the individual values for these variables). Maybe I’m going to have to test cases?
Read: Can’t do much with the question stem, besides writing down that info.
Jot:
How should I approach this?
Hmm. They’re all positive integers greater than 1. That’s intriguing; why greater than 1 and not the more common greater than 0? I’ll need to think about that.
I just have to find the sum of the variables, not the individual variables. And each statement uses all three variables and provides some real numbers.
So, the question is whether I can rearrange that info somehow to tell me the sum, even if it doesn’t tell me the individual variables. Let’s see.
“(1) xyz = 70”
If they’re all integers, then they have to be made up of the various possible factors of 70.
Oh! This is key: they’re all integers greater than 1, so I can ignore the factor pair (1, 70). In other words, what I really care about is the prime factors of 70. I was wondering why they told me such a weird piece of info.
Okay, so I need to break 70 down into its prime factors and then test cases with those numbers to see whether I get a definitive sum or multiple sums.
70 = (7)(10) = (7)(2)(5)
There are three variables…and three prime factors. So the three variables have to be 7, 2, and 5! The sum of those three numbers is always the same, regardless of the order in which the addition occurs. Statement (1) is sufficient to answer the question.
“(2)
(Remember, reflect first! Don’t just dive in.) Fractions are annoying, so I could try cross-multiplying. That would give me 10x = 7yz. That doesn’t actually look simpler (at least, not to me!), though.
Oh, or how about this: x could be 7 and yz could be 10, in which case y and z have to be 2 and 5, in some order. That works! And those are the same numbers as in statement (1). Yay!
But wait. Reflect some more. This is one possible solution, yes, but is it the only one?
What if x = 14 and yz = 20? In that case, statement (2) is still true, but the values have changed. Will the sum be the same? No! It’ll be larger, since the values are larger.
Statement (2) allows more than one possible sum, so it is not sufficient to answer the question.
The correct answer is (A).
Now, are you starting to see how the GMAT has found a way to make NP hard? The pure math that had to be done here wasn’t crazy hard. But the conceptual thinking was definitely not what we were taught in school. This is how they’re going to test your adult-level thinking of NP topics.
Key Takeaways for Divisibility and Primes on the GMAT:
(1) They’re not testing pure math here. They’re testing theory. You’re going to have to learn how to take Number Properties rules and think about them conceptually.
(2) In order to do that, you’ll need to start picking up on the clues that they give in the way that they present the information. One key clue was that integers greater than 1 piece coupled with multiplication later in the problem (in the statements). When you multiply integers greater than 1, those integers become factors of the larger number you create. This is your clue that you’re being asked about the factors of some number—and this puts you squarely in the category of Divisibility and Primes, one of the main topic areas under Number Properties. ???
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* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.
Keep Reading: Part 2
Stacey Koprince is a Manhattan Prep instructor based in Montreal, Canada and Los Angeles, California. Stacey has been teaching the GMAT, GRE, and LSAT for more than 15 years and is one of the most well-known instructors in the industry. Stacey loves to teach and is absolutely fascinated by standardized tests. Check out Stacey’s upcoming GMAT courses here. As always, you can try your first class for free!