The Correct Way to Factor on the GMAT

One of the most important concepts on the GMAT quant section is the notion of factors. Because there is no calculator on the exam, the multiplications and divisions tend to heed integer numbers. Dividing 100 by 2 might be trivial, but dividing 1100 by 22 might hinge on your recognition of the common factor of 11 to avoid tedious and time-consuming calculations.

One important aspect worth mentioning about factors is that prime factors are the best way to describe any number. As an example, 100 can be factored out to be 10×10, or it could be factored out to be 4×25. Or 2×50, or 2x2x5x5. Which factoring is correct? In a way, they all are. (In another, more accurate, way, 2x2x5x5 is correct). Why would you pick 2^2 x 5^2? Because it’s only prime numbers. Primes provide the easiest way to designate the number and will be consistent for all numbers independently of factoring method. This is akin to saying 2+2 is 4 and not 8/2 or 1024/256. Sure the other answers are technically correct but they are somewhat arbitrary and borderline deceiving.

A common way of testing factors on the GMAT is by giving abstract numbers and asking how they can be divided. As an example, consider for the following GMAT question:

What is the greatest power that 5 can be raised to so that the resulting number is a factor of 15!

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Looking at 15! (pronounced 15 factorial, not 15 emphatically entering the room), we probably won’t spend the time on test day to calculate that it can be rewritten as 1,307,674,368,000. However I already knew that it would have to end in exactly three 0’s. Why? Because 0 is only formed by multiplying 2 by 5. Thus every zero will indicate the presence of both a 5 and 2. The three 0’s (like the three tenors, but less successful) are the result of multiplying 2’s and 5’s in the prime factors of these numbers. Unsurprisingly, there will be a lot of 2’s in this product, but only a handful of 5’s.

How many 5’s exactly? Well 5 is a prime number, so it won’t be formed by the product of two different factors. It will only appear in numbers that have a five in them, or the numbers that end in 5 or 0. In 15! there are three numbers that will have a 5 in them: 5, 10 and 15. Thus we know that this number is the product of 5^3 as well as several other prime numbers.

Solving this problem does not require us to take the full prime factorization of 15x14x13x12x11x10x9x8x7x6x5x4x3x2x1, but you certainly could do it if you wanted to. The answer is 211 x 36 x 53 x 72 x 111 x 131. (Fun fact, this number will have exactly 4032 factors… collect them all!). The only number we are interested in is that 53, which means that we can raise 5 to the third power and still get a factor of 15 factorial. Thus, 54 will not qualify. Answer choice B is correct.

If that made sense, let’s look at a similar problem with another classic GMAT hook.

If m is the product of all integers from 1 to 40, inclusive, what is the greatest integer p for which 10^p is a factor of m?

(A) 4
(B) 7
(C) 8
(D) 9
(E) 11

If you understood the explanation above, you’ll realize that 10p means how many 2×5’s will be present in this product. The 2’s won’t be the limiting factors (just 32x16x8 gives us more 2’s than any of the answer choices are offering). The number of 5’s will therefore limit the number of 0’s the product will end with. How many 5’s are there in 40! (a number so long writing it out here seems silly). Well which numbers end with 5 or 0? 5, 10, 15, 20, 25, 30, 35 and 40.

These are the eight numbers that have 5 as a factor. This looks like answer choice C. In fact, the majority of people who see this question instinctively gravitate towards C. Much like a traveler parched in the desert, the C is a mirage. (*cough*). These eight numbers are undoubtedly the only ones with 5 as a factor, but one of them is still deceptive. The number 25 actually contains two of them. (Much like the Van Damn Classic film Double Impact). The total number of 5’s among the factors will actually be 9, and not 8. The correct answer is thus D.

Even if you don’t encounter a question that explicitly asks you for factors on the GMAT, the ability to break down numbers into their prime components is a huge advantage on test day. Being at ease with factoring can provide you the answer on certain questions and help you save time on others. In your journey to maximize your GMAT grade, your proficiency with this skill may very well prove to be your X-factor.

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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.