2. Build pattern recognition skills
Successful students become adept at pattern recognition, the ability to see order within a chaotic environment. Pattern recognition becomes much easier if you are able to identify common patterns. For example, one common pattern comes from a quadratic identity in the form of (x + y) (x – y) = x2 – y2, known as the “difference of squares.” Are any of the following in the form of the difference of squares?
(1) (555)2 – (55)2 (2) x2 – 1 (3) x100 – y100
Most see that (555)2 – (55)2 is in the form of a difference of squares, because it is “something squared” minus “something squared.” What they fail to see is that x2 – 1 and x100 – y100 are as well.
x2 – 1 is a difference of squares, since x2 is the square of x and 1 is the square of 1 (12 = 1). Thus, x2 – 1 = x2 – 12 = (x + 1)(x – 1).
x100 – y100 is also a difference of squares, since x100 is the square of x50 and y100 is the square of y50.
In other words, x100 – y100 = (x50)2 – (y50)2 = (x50 – y50)(x50 + y50).
If you didn’t see that x2 – 1 and x100 – y100 were in the form of a difference of squares, you likely had a hard time seeing the pattern that something in the form of x2 – y2 is a difference of squares. The better you become at recognizing patterns (like the difference of squares), the better you’ll perform.
Try the following problems. In each, attempt to spot the difference of square patterns.
Even if you haven’t done well with math in the past, there are many simple patterns found on the GMAT that you can learn to recognize. I see students, many without strong math backgrounds, build these patterns all the time, and I see how this improves their scores. Put in the necessary effort, and you’ll be amazed at what you can learn to recognize.
3. Become familiar with the not-so-familiar
GMAT questions are designed to be fair and equally accessible to all students applying to business school. However, concepts are often tested in non-standard ways. For example, even my students who are engineers and finance quants tend to answer the following question incorrectly, despite its simplicity and their strong background in math:
4x + 4x + 4x + 4x =
A. 44+x B. 44x C. 164x D. 4x+1 E. 164+x
Many choose answer B – a trap – because it just “looks correct” and appeals to the intuition of even skilled math students. Most who choose B somehow mistake this problem for a more familiar one involving multiplication [(4x) (4x)(4x)(4x) = 44x] but the question above involves addition. Adding the four terms yields 4x + 4x + 4x + 4x = (4)(4x) ⇒ 414x = 4x+1, answer choice D.
Another type of problem you may never have studied in your math classes is “units digit” problems.
What is the units digit of 486 ?
A. 0 B. 2 C. 4 D. 6 E. 7
This problem combines the “unfamiliar” with “pattern recognition skills.” Even with a powerful calculator, it’s hard to determine the answer. Therefore, you might be tempted to choose answer choice A. Another naïve GMAT test taker might choose C, because the problem encourages thinking of many 4s. Actually, the correct answer is D.
Here’s a quick explanation of the rationale. Think about how powers of 4 are generated: 41 = 4, 42 = 16, 43 = 64, 44 = 256. Do you see a pattern emerging in the units digit of these answers? Note that if 4 is raised to an odd power, the units digit is 4, and if 4 is raised to an even power, then the units digit is 6.
This is the pattern that you will use to answer the question. The question asks what the units digit is when 4 is raised to the 86th power. Because 86 is an even number, the units digit of 486 will, therefore, be 6, answer choice D. With the right approach you have the correct answer in less than 30 seconds, with no need for a calculator! On test day, while other students are still struggling with this problem, you will have already answered it correctly and will have moved on to the next question. You’ll have a two-fold boost to your confidence: you answered it correctly, and you now have the luxury of extra time for answering the truly difficult questions that lie ahead.
Your opportunity is to become familiar with the unfamiliar way GMAT questions are structured. Avoid the temptation to analyze these questions in the same ways you analyzed questions in your algebra classes.
4. Watch for unhelpful cognitive patterns
In the last two articles, we examined the importance of careful awareness. It’s imperative that you not take the bait offered by some answer choices, particularly on data sufficiency questions.
Try the following question:
If n is an integer, is (n3 – n)/x an integer?
1) x = 6
2) n = 2
Many students choose the easiest deduction: With the values of both x and n, we can answer the question. Therefore, most choose C, which happens to be incorrect. Why? Likely, it’s because they performed a shallow analysis of the problem and allowed lazy thinking to dominate the thought process. The answer is actually A.
Let’s begin with the content issue. A student must first recognize that the numerator given in the stem can be factored. The problem seems a little easier, once we perform the following step:
(n3 – n)/x = n(n2 – 1)/x
At this point, the student must recognize that n2 – 1 represents a difference of squares: x2 – y2 = (x – y)(x + y). Thus, we can further factor the numerator as:
n(n2 – 1)/x = n(n – 1)(n + 1)/x
Now, the student must recognize an important pattern in the numerator: n(n–1)(n+1) represents the product of three consecutive integers. It’s easier to see this pattern, once we reorder the terms as (n–1), n, and (n+1).
So, we finally have restructured the question to be:
If n is an integer, is (n-1)(n)(n+1)/x an integer?
It is an obscure fact that the product of any three consecutive integers is a multiple of 6. This fact may be obscure, but it’s based on a simple analysis: with any three consecutive integers, at least one is always a multiple of 2, and one is always a multiple of 3. With this knowledge, we can see that statement 1, alone, is sufficient. So, the answer is A.
On GMAT quant problems – and especially data sufficiency questions – you must resist the impulse to run with the easiest deduction possible. Instead, perform a deeper analysis of the question. Learning such a rigorous process of analysis takes a little time, but you can learn it and practice it while you study, and then successfully use it on test day.
The theme of my “GMAT Unlocked” series of articles is that you can outperform your peers, not so much by how much smarter you are, but rather by how much smarter and harder you have prepared for the GMAT. In this installment, I’ve focused on the subtle and not-so-subtle approaches and techniques for solving a variety of quantitative problem types that you are sure to see on the GMAT. You now have additional weapons in your arsenal that can be effectively deployed to yield a GMAT victory! Don’t give up. Study hard. Study smart. You deserve a strong GMAT score. You can do this! Don’t let anyone tell you that you can’t.
Scott Woodbury-Stewart is Founder and CEO of Target Test Prep, one of the fastest growing GMAT test prep firms on the market. Scott is writing a special four-part series for Poets&Quants with advice for the GMAT. He and his team can be contacted for a personal consultation.