It’s All Greek to Me: How to Use Greek Concepts to Beat the GMAT

Aero_img084The ancient Greeks were, to put it mildly, really neat. They created or helped to create the foundations of philosophy, theater, science, democracy, and mathematics – no small accomplishment for a small war-torn civilization from over two millennia ago. Many of our contemporary ideas, beliefs, and traditions are rooted in contributions made by Greek thinkers, and the GMAT is no exception.

A few months ago, I wrote about this difficult Data Sufficiency question.

When I first encountered this problem I couldn’t help but wonder what kind of mad scientist question-writer engineered it. Where would such an idea even come from? It turns out, it wasn’t a GMAC employee at all, but Archimedes, the famous Greek geometer and coiner of the phrase “Eureka!”

The question is based on his attempt to trisect an angle with only a straight edge and a compass. (Alas, Archimedes’ work, though ingenious, was not technically a correct solution to the problem, as it provides only an approximation.) The reader is hereby invited to contemplate the kind of person who encounters a proof by Archimedes and instinctively thinks, “This would make an excellent Data Sufficiency question on the GMAT!” We’d like to believe that the good folks at GMAC are just like you and me, but perhaps not.

So this got me thinking: what other interesting Greek contributions to mathematics might be helpful in analyzing GMAT questions? In Euclid’s work Elements, he offers a simple and elegant proof for why there is no largest prime number. The proof proceeds by positing a hypothetical largest prime number “p.” We can then construct a product that consists of every prime number 2*3*5*7….*p. We’ll call this product “q.”

The next consecutive number will be q + 1. Now, we know that “q” contains 2 as a factor, as “q,” supposedly, contains every prime as a factor. Therefore q +1 will not contain 2 as a factor. (The next number to contain 2 as a factor will be q + 2.) We know that “q” contains 3 as a factor. Therefore q + 1 will not contain 3 as a factor. (The next number to contain 3 as a factor will be q + 3.)

Uh oh. If “p” really is the largest prime number, we’ve got a problem, because q + 1 will not contain any of the primes between 2 and p as factors. So either q + 1 is itself prime, or there is some prime greater than p and less than q + 1 that we’ve failed to consider. Either way, we’ve proven that p can’t be the largest prime number – I told you the Greeks were neat.

One axiom that’s worth internalizing from Euclid’s proof is the notion that two consecutive numbers cannot have any factors in common aside from 1.  When q contains every prime from 2 to p as a factor, q + 1 contains none of those primes. How would this be helpful on the GMAT? Glad you asked. Check out this question:

x is the product of all even numbers from 2 to 50, inclusive. The smallest prime factor of x + 1 must be:

(A) Between 1 and 10

(B) Between 11 and 15

(C) Between 15 and 20

(D) Between 20 and 25

(E) Greater than 25

We’re given information about x, and we’re asked about x + 1. If x is the product of all even numbers from 2 to 50, we can write x = 2 * 4 * 6 …* 50. This is the same as (1*2) * (2*2) * (3*2)… (25*2), which means the product consists of all the integers from 1 to 25, inclusive, and a bunch of 2’s.

So now we know that every prime number between 2 and 25 will be a factor of x. What about x + 1? (Paging Euclid!) We know that 2 is not a factor of x + 1, as 2 is a factor of x, and so the next multiple of 2 would be x + 2. We know that 3 is also not a factor of x + 1, as 3 is a factor of x, and so the next multiple of 3 would be x + 3. And once we’ve internalized that two consecutive numbers cannot have any factors in common aside from 1, we know that if all the primes between 2 and 25 are factors of x, none of those primes can be factors of x + 1, meaning that the smallest prime of x, whatever is, will be greater than 25. The answer, therefore, is E.

Takeaway: One of the beautiful things about mathematics is that fundamental truths do not change over time. What worked for the Greeks will work for us. The same axioms that allowed ancient mathematicians to grapple with problems two millennia ago will allow us to unravel the toughest GMAT questions. Learning a few of these axioms is not only interesting – though I’d caution against bringing up Archimedes’ trisection proof at a dinner party – but also helpful on the GMAT.

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By David Goldstein, a Veritas Prep GMAT instructor based in Boston. You can find more articles by him here.

GMAT Tip of the Week

GMAT prepA Trick That Might Factor In

(This is one of a series of GMAT tips that we offer on our blog.)

The quantitative section of the GMAT has a known emphasis on factorization of numbers, asking a variety of questions about divisibility, primes, least common multiple, etc. One fairly common question type asks, “How many unique factors does (number) have?”

While there are certainly strategies to answer this question efficiently without any kind of shortcut trick, a pretty slick shortcut does exist:

The first step would likely be your first step in any factorization problem; break the number down in to its prime factors. Take a number like 24, which breaks in to the factors 2*2*2*3.

Secondly, express that factorization as the product of exponents. In this case, it’s 2^3 * 3^1.

Next, discard the bases, and add one to each of the exponents. Here we’d have 3 and 1, and add one to each to make them 4 and 2.

Then, multiply the exponents-plus-one. 4*2 is 8, and there are 8 unique factors of 24:

1, 2, 3, 4, 6, 8, 12, and 24

The proof is a bit messy, but derives from another GMAT concept — combinatorics — which relies heavily on the use of, of all things, factorials.

If you blank on a trick like this, however, note that you can systematically come up with the above list of factors on your own, by taking the prime factors (2, 2, 2, and 3) and 1 (a factor of any integer), and multiplying each possible combination of them:

1
1*2 = 2
1 * 3 = 3
2*2 = 4
2 * 2 * 2 = 8
2 * 3 = 6
2 * 2 * 3 = 12
2 * 2 * 2 * 3 = 24

It’s a bit more time consuming, but doesn’t require that you memorize a trick precisely in order to solve the problem. As with all shortcuts on the GMAT, if they “click” for you, they’ll save you time, which then you can use for those problems that simply require more thought. Please don’t spend all of your time memorizing tricks, as the GMAT is written to reward “higher order thinking”. That said, a time-saving trick or two can provide you with an additional few minutes on the exam, and the corresponding confidence that you’re primed to post a high score.

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