When going through the quantitative section of the GMAT, you will often be confronted by numbers that are, shall we say, unwieldy (some people refer to them as “insane”). It is common on the exam to see numbers like 11!, 15^8, or even 230,050,672. Regardless of the form of the number, the common mistake that many novice test takers make is the same: They try to actually solve the number.
Now, some numbers are spelled out down to the decimals, but other numbers, such as 11!, seem unnecessarily abstract. 11 factorial is a big number, but wouldn’t it be simpler if I had a concrete number in front of me instead of a shorthand notation for 10 multiplications. The answer is: not really. If you wanted to expand 11! To get a longhand answer, you’ll end up with a large concrete number that is no easier to manipulate than the shorthand you had before. For example, 11! is actually 39,916,800. Does that make it any easier to use? Again, the answer is: not really.
In essence, every time you see a big number like this, the GMAT is baiting you into performing tedious calculations that don’t help you in any way. Having a cumbersome number is the GMAT’s way of saying “Don’t try and solve this with brute force, there’s a concept here you should recognize”. While it’s uncommon for the GMAT to actually speak, given that it’s an admissions exam, it actually is telling you loud and clear that concentrating on the number is a trap. There will always be some element that will help highlight the underlying issue without performing tedious math.
There are many concepts that may come into play, and it’s hard to approach these questions with a single standard approach, but some elements repeat more frequently than others. One of the first things to look for is the units digit. The units digit gives away many properties of a number. As an example, 39,916,800 ends with a 0, indicating that it is even, and that it is divisible by 10. Different units digits can yield different number properties, so you can learn a lot from one simple digit. The factors of the number in question can often unlock clues as to which numbers to look for among the answer choices. Finally the order of magnitude can also play a pivotal role in determining how to approach a question.
Since we don’t have one definitive strategy, let’s test our mental agility on an actual GMAT question:
For integers x, y and z, if ((2^x)^(y))^(z) = 131,072, which of the following must be true:
(A) The product xyz is even
(B) The product xyz is odd
(C) The product xy is even
(D) The product yz is prime
(E) The product yz is positive
This question is significantly easier if you recognize which power of two 131,072 is off the bat (I knew that Computer Science degree would be good for something). However, let’s approach this knowing that 131,072 is a multiple of two, but that calculating which one would require more time than the two minutes we have earmarked for this question. Furthermore, simply knowing that 131,072 is a power of 2 gives us all the information we really need to solve this question.
We know x, y and z will combine to form some integer, but we’re not sure which. Let’s call it integer R (as in Ron) for simplicity’s sake. Moreover, the way the equation is set up, the powers will all be multiplied by one another, meaning that their exact order won’t matter. As such, the commutative law of mathematics confirms that if ((2^5)^(3))^(2) is the exact same thing as ((2^3)^(2))^(5). If the order doesn’t matter, then there are a lot of potential situations that could occur. So R will equal x + y + z, but the order won’t change anything. Let’s look at the answer choices, and start from the end because they’re easier to eliminate.
Answer choice E asks us whether y*z must be positive. If y*z gives us some positive number, then x would just be whatever is left over to form R. It doesn’t matter is y*z is positive or negative, as x can just come and make up the difference. Let’s say y*z = 4, then x would just be R – 4. If, instead, y*z = -4, then x would just be R – 12 and there would be no difference. In other words, as long as one variable is unrestricted, it will always be able to make up for the restriction on the other two. If you recognize this, you can eliminate C, D and E for the same reason. Two out of three ain’t bad, but in this case, it ain’t enough.
This brings us down to answer choices A and B, which are complimentary. Either the product of the three numbers is even, or it is odd. One of these, logically, must be true. Unfortunately, the best way to verify this appears to be doing the calculation longhand (like the petals of a flower: she loves me, she loves me not). Herein lays a potential shortcut: the units digit. Since the number is a power of two, we can simply follow the pattern of multiples of two and see what we get. Considering primarily the units digit (underlined for emphasis):
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
You probably don’t have to go this far to notice the pattern, but it doesn’t hurt to confirm if you’re not sure after 2^5. Essentially, the unit digit oscillates in a fixed pattern: 2, 4, 8, 6, and then repeats. This is helpful, because the number in question ends with a 2, and every power of two that ends with a 2 is either 2^1, 2^5, 2^9, etc. All of these numbers are odd powers of 2, repeating every fourth element. With this pattern clearly laid out, it becomes apparent that the answer must be that the product of these three variables must be odd. As such, answer choice B is correct here. We can also probably deduce from order of magnitude that 131,072 is 2^17.
When it comes to large numbers on the GMAT, you should never try to use brute force to solve the problem. The numbers are arbitrarily large to dissuade you from trying to actually calculate the numbers, and they can be made arbitrarily larger on the next question to waste even more of your time. The GMAT is a test of how you think, so thinking in terms of constantly calculating the same numbers over and over limits you to being an ineffective calculator. Your smart phone currently has at least 100 times your computational power (but not the ability to use it independently… yet…). Brute force may break some doors down, but mental agility is a skeleton key.
Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.