But You and I, We’re Just a Couple of Squares… What Difference Could We Possibly Make…?

The savvy GMAT-goer knows that the work on a Problem Solving question is best undertaken only after a survey of the answer choices makes clear just how much work — and what type of work — is really necessary. For instance, a 160/1600/16000/… set of choices tells you you can focus all your care on the magnitude of your answer; a 16000/25000/36000/… set of choices tells you you can forget about all those trailing 0’s and just focus on the “head” of the answer. As we stress in our Foundations of GMAT Logic book, *the answer choices are part of the problem*.

BUT the answer choices’ power is that they help you direct your work efficiently; they should *not*, on the contrary, become your hang-ups. Nowhere is this admonition more pertinent than on questions that query potential rather than “is”ness, smuggling a sort of “check all that apply” task in via some sleight of test craftsmanship. Which is to say — everybody loves those Roman numeral problems… right?

Let’s look at an example:

If x and y are squares of integers, which of the following could be the value of y – x?

I. 20

II. 46

III. 73

A) I only

B) III only

C) I and II only

D) I and III only

E) I, II, and II

The thing to keep in mind on “could be” questions is that the makers of the GMAT are *not* sadists who delight in making you jump through arbitrary hoops; they’re much more along the lines of benevolent architects who provide you with an arena in which to demonstrate the kind of thinking you’re capable of. If you’ve pegged them as sadists, you’re indeed quite likely to approach the test by attempting to jump through arbitrary hoops — a tactic that will most likely be frustrating and not particularly successful. If, in contrast, you adopt the benevolent-architect view, you’re far more apt to approach each problem tuned into what talent it’s offering you the chance to exhibit.

So, how do we crack a problem like the above? You’ve got these shiny, specific candidates for values… should you just have a go at it, and see, for instance, whether you can come up with two squares that differ by 20? And repeat for 46, and repeat for 73? Mmmm… not recommended[1] — this could prove a time sink of unknowable depth. A good way to know this is a bad idea is that there is no good reason in the world that a test should want to ascertain whether you can feel your way to two perfect squares that differ by 20, since the answer to that question would say nothing about anything but–well–your ability to feel your way to two perfect squares that differ by 20. (Quite frankly, you deserve the chance to flaunt more than that.)

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Okay. If we don’t go chasing them individually, how *do* we use these values? The key is to think of them as members representing larger classes of numbers. For instance, 73 could conceivably have been sent here on behalf of all primes, or still more simply, on behalf of all odd numbers. Likewise, 20 and 46 could both have been sent as representatives of the whole even number population, but it’s quite likely there’s a salient difference between them, too. For instance, 20 might represent a more “elite” class of evens. Even numbers, by definition, are precisely those numbers with the property that you can chop them in half and still wind up with an integer, but in comparison to numbers in 46’s class, numbers in 20’s class have evenness to spare, as it were, since you can chop them in half and even still wind up even. Less glamorously put: all (and only) multiples of 4 have this property — “backup” evenness.

Armed with these hypotheses, let’s do some actual algebra. First, *use* *the fact that x and y are perfect squares!* You can be pretty confident that this piece of information was given to you for a reason, so it’s a good idea to write x in terms that incorporate it. Let’s let x = k^{2 }for some integer k. Then y might be anything, so why not start small; we’ll increment k by 1 to make x and y consecutive squares, where y = (k+1)^{2}. Then y – x = (k+1)^{2} – k^{2} = (k^{2} + 2k + 1) – k^{2} = 2k + 1. This is very good news! Remember, k can take on any integer value at all, so we’ve just shown that every single possible 2k + 1 — that is, **every single possible odd number** — is the difference of some two perfect squares. So 73, we know you’re safe for sure.

It would be silly to stop here, since you should already be on alert for some possible even number antics. So what happens if we keep x as k^{2}, but increment k by 2 now so that y = (k+2)^{2}? Then y – x = (k+2)^{2} – k^{2} = (k^{2} + 4k + 4) – k^{2} = 4k + 4. But again, remember, k can take on any integer value at all, so we’ve just shown that every single possible 4k + 4 — that is, **every single possible multiple of 4 **– is the difference of some two perfect squares. So 20, you’re in.

What about 46? Is there a way to generate those just-the-basics, nothing-to-spare evens? Your instinct here should be ::no::, because if incrementing k by 2 already guaranteed differences that were multiples of 4, incrementing k by an even number should do the same. You can convince yourself of this: if y = (k + any even)^{2}, then y = k + 2a for some integer a. Then y – x = (k + 2a)^{2} – k^{2} = k^{2} + 2(2a)(k) + (2a)^{2} – k^{2} = 4ak + 4a^{2} … which is most certainly a multiple of 4. Subtract an even square from another even square, or an odd square from another odd square, and you can get any multiple of 4 you want, but nothing else. So sorry, 46, three’s a crowd and you’re impossible.

If x and y are squares of integers, which of the following could be the value of y – x?

I. 20

II. 46

III. 73

A) I only

B) III only

C) I and II only

**D) I and III only
**E) I, II, and II

[1] Which is not to say impossible. Except in the case for which it is!

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*Ashley Newman-Owens is a top Veritas Prep GMAT instructor based in Boston. After graduating with a BA in English Lit, she received an MFA in Creative Nonfiction, and has now gone onto pursuing a degree in Math Education at Tufts, where she is exploring the nature of how young students understand functions.*