Quarter Wit Quarter Wisdom: What is Your Favorite Number?

Quarter Wit, Quarter WisdomFans of The Big Bang Theory will remember Sheldon Cooper’s quote from an old episode on his favorite number:

The best number is 73. Why? 73 is the 21st prime number. Its mirror, 37, is the 12th and its mirror, 21, is the product of multiplying 7 and 3… and in binary 73 is a palindrome, 1001001, which backwards is 1001001.”

Though Sheldon’s logic is infallible, my favorite number is 1001 because it has a special role in standardized tests.

1001 is 1 more than 1000 and hence, is sometimes split as (1000 + 1). It sometimes appears in the a^2 – b^2 format such as 1001^2 – 1, and its factors are 7, 11 and 13 (not the factors we usually work with).

Due to its unusual factors and its convenient location (right next to 1000), it could be a part of some tough-looking GMAT questions and should be remembered as a “special” number. Let’s look at a question to understand how to work with this  number.

Which of the following is a factor of 1001^(32) – 1 ?

(A) 768
(B) 819
(C) 826
(D) 858
(E) 924

Note that 1001 is raised to the power 32. This is not an exponent we can easily handle. If  we try to use a binomial here and split 1001 into (1000 + 1), all we will achieve is that upon expanding the given expression, 1 will be cancelled out by -1 and all other terms will have 1000 in common. None of the answer choices are factors of 1000, however, so we must look for some other factor of 1001^(32) – 1.

Without a calculator, it is not possible for us to find the factors of 1001^(32) – 1, but we do know the prime factors of 1001 and hence, the prime factors of 1001^32. We may not be able to say which numbers are factors of 1001^(32) – 1, but we will be able to say which numbers are certainly not factors of this!

Let me explain:

1001 = 7 * 11 * 13 (Try dividing 1001 by 7 and you’ll get 143. 143 is divisible by 11, giving you 13.)

1001^32 = 7^32 * 11^32 * 13^32

Now, what can we say about the prime factors of 1001^(32) – 1? Whatever they are, they are certainly not 7, 11 or 13 – two consecutive integers cannot have any common prime factor (discussed here and continued here).

Now look at the answer choices and try dividing each by 7:

(A) 768 – Not divisible by 7

(B) 819 – Divisible by 7

(C) 826 – Divisible by 7

(D) 858 – Not divisible by 7

(E) 924 – Divisible by 7

Options B, C and E are eliminated. They certainly cannot be factors of 1001^(32) – 1 since they have 7 as a prime factor, and we know 1001^(32) – 1 cannot have 7 as a prime factor.

Now try dividing the remaining options by 11:

(A) 768 – Not divisible by 11

(D) 858 – Divisible by 11

D can also be eliminated now because it has 11 as a factor. By process of elimination, the answer is A; it must be a factor of 1001^(32) – 1.

I hope you see how easily we used the factors of 1001 to help us solve this difficult-looking question. And yes, another attractive feature of 1001 – it is a palindrome in the decimal representation itself!

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Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

How to Approach Difficult GMAT Problems

SAT/ACTMy students have a hard time understanding what makes a difficult GMAT question difficult. They assume that the tougher questions are either testing something they don’t know, or that these problems involve a dizzying level of complexity that requires an algebraic proficiency that’s simply beyond them.

One of my main goals in teaching a class is to persuade everyone that this is not, in fact, how hard questions work on this test. Hard questions don’t ask you do to something you don’t know how to do. Rather, they’re cleverly designed to provoke an anxiety response that makes it difficult to realize that you do know exactly how to solve the problem.

Take this official question, for example:

Let a, b, c and d be nonzero real numbers. If the quadratic equation ax(cx + d) = -b(cx +d) is solved for x, which of the following is a possible ratio of the 2 solutions?

A) –ab/cd
B) –ac/bd
C) –ad/bc
D) ab/cd
E) ad/bc

Most students see this and panic. Often, they’ll start by multiplying out the left side of the equation, see that the expression is horrible (acx^2 + adx), and take this as evidence that this question is beyond their skill level. And, of course, the question was designed to elicit precisely this response. So when I do this problem in class, I always start by telling my students, much to their surprise, that every one of them already knows how to do this. They’ve just succumbed to the question writer’s attempt to convince them otherwise.

So let’s start simple. I’ll write the following on the board: xy = 0. Then I’ll ask what we know about x or y. And my students shrug and say x or y (or both) is equal to 0. They’ll also wonder what on earth such a simple identity has to do with the algebraic mess of the question they’d been struggling with.

I’ll then write this: zx + zy = 0. Again, I’ll ask what we know about the variables. Most will quickly see that we can factor out a “z” and get z(x+y) = 0. And again, applying the same logic, we see that one of the two components of the product must equal zero – either z = 0 or x + y = 0.

Next, I’ll ask if they would approach the problem any differently if I’d given them zx = -zy – they wouldn’t.

Now it clicks. We can take our initial equation in the aforementioned problem: ax(cx +d) = -b(cx+d), and see that we have a ‘cx + d’ on both sides of the equation, just as we’d had a “z” on both sides of the previous example. If I’m able to get everything on one side of the equation, I can factor out the common term.

Now ax(cx +d) = -b(cx+d) becomes ax(cx +d) + b(cx+d) = 0.

Just as we factored out a “z” in the previous example, we can factor out “cx + d” in this one.

Now we have (cx + d)(ax + b) = 0.

Again, if we multiply two expressions to get a product of zero, we know that at least one of those expressions must equal 0. Either cx + d = 0 or ax + b = 0.

If cx + d = 0, then x = -d/c.

If ax + b = 0, then x = -b/a.

Therefore, our two possible solutions for x are –d/c and –b/a. So, the ratio of the two would simply be (-d/c)/(-b/a). Recall that dividing by a fraction is the equivalent of multiplying by the reciprocal, so we’re ultimately solving for (-d/c)(-a/b). Multiplying two negatives gives us a positive, and we end up with da/cb, which is equivalent to answer choice E.

Takeaway: Anytime you see something on the GMAT that you think you don’t know how to do, remind yourself that the question was designed to create this false impression. You know how to do it – don’t hesitate to dive in and search for how to apply this knowledge.

Plan on taking the GMAT soon? We have GMAT prep courses starting all the time. And be sure to follow us on FacebookYouTubeGoogle+ and Twitter!

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

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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