Number Properties

On the face of it, the topic seems straightforward: You 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 CAT 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. 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. “*If x, y, and z are integers greater than 1, what is the value of x + y + z? “(1) xyz = 70 “(2) x/yz=7/10”   Glance: DS. Three variables. The question is a combo (that is, you don’t necessarily have to find the individual values for these variables). You are going to have to test cases. Read: Can’t do much with the question stem, besides writing down that info.     They’re all positive integers greater than 1. That’s intriguing; why greater than 1 and not the more common greater than 0? You 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 you can rearrange that info somehow to tell you the sum, even if it doesn’t tell you 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. This is key: they’re all integers greater than 1, so you can ignore the factor pair (1, 70). In other words, what you really care about is the prime factors of 70. Okay, so you 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 variable 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) x/yz=7/10” 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). 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 CAT 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 CAT: (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.