ROn Point: Divisor Problems on the GMAT

On the GMAT, you will face a variety of questions that you can prepare for. Not to be an auctioneer, the section boasts arithmetic problems, factor problems, algebra problems, geometry problems, stats problems, probability problems, data sufficiency problems, work rate problems, ratio problems, even combinatorics problems. However on the quant section you can often run into an unfamiliar question type that can reasoned out with some basics of algebra and clear conceptual thinking. When faced with this type of outlandish question, you only have one basic directive: solve it.

 

If a and b are positive integers such that a/b = 6.65, which of the following must be a divisor of a?

a) 3

b) 11

c) 19

d) 38

e) 40

This type of question doesn’t show up very often on the GMAT, but when it does, it may leave you scratching your head. If the division of a and b had left an easily identifiable fraction, such as 0.25 or 0.10, then this might have lead you to think about specific fractions in an effort to solve this question. The value 0.65 is the equivalent of 13/20, as many students hoping to have passed a high school exam may remember. No matter what the fraction, though, the path to success always looks the same on these questions.

To elaborate, consider that what a/b = x really means. This means that if I divide a into b parts, I get x portions. In tangible terms, if I have 40 cookies, and 5 students, each student gets 8 cookies. If a new student comes in (and messes up the perfectly round numbers I purposely chose), I now have 6 students for my 40 cookies, and each student gets 7.28 cookies, or 7 2/7 cookies, or possibly 7 cookies with remainder 2 ( and probably a stomach ache). It’s clear that if I multiply the number of cookies by the number of students, I get the initial number of cookies I split up. In math terms, this means if I multiply x by b, I will get a. Since the question is looking for divisors of a, let’s go ahead and do that in our initial problem.

a/b = 6.65

Applying the math to isolate a (multiply both sides by b)

a = 6.65 * b

Multiplying by 100 to remove fractions leaves us with:

100 * a = 665 * b

Now we want to be able to say something about a. So let us divide both sides by 5, which isn’t trivial without a calculator, but you can think of 665 as 500 + 165, and 500/5 = 100. You can repeat this trick for 165 = 100 + 65, where 100/5=20.  65/5 = 13, so in the end you have 133. This means that

20 * a = 133 *b

What does this tell us? It indicates that 20 * a must be exactly equal to 133 * b. This means that 133 must be a divisor of 20 * a, and since 133 is not a divisor of 20, it must be a divisor of a. This may not be obvious at first, but it is the key to the entire question, or at least of the second act. The third act will be about determining the factors of 133. (also determining why Iago betrayed Othello, time permitting).

Now that we know that 133 must be a factor of a, we must either factor 133 and compare the factors to the answer choices, or else divide 133 by each answer choice to see if any give an integer. As an aside, there must be exactly one that gives an integer; zero or two answers would indicate that a mistake was made along the way. At that point the math should be double-checked or a strategic guess should be made, depending on the time remaining. Going through each choice quickly:

3: 1+3+3 = 7, so 133 is not divisible by 3.

11: 133-110 = 23. 23 is not divisibly by 11. Those who know the multiplication table readily will also recognize that 132 is 11×12, and thus 133 is not divisible by 11.

19: If we keep adding 19s we get to 95, and then 114, and then 133. Conversely, you might see that 133 is divisible by 7, leaving a quotient of 19.

38: 133 is odd and no matter how many times you multiply an even number, it remains even.

40: Same as 38, but even faster.

The Veritas Prep course highlights that factoring can be a major element on the GMAT and is possibly the most important skill you can learn during your preparation. This is demonstrated in this question that needs to be solved in three acts, at least two of which require a good understanding of factorization. It is undeniable that this skill will be a major factor in your success on test day.

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Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you occasional tips and tricks 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.

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