GRE Math for People Who Hate Math: A Gentle Introduction to GRE Divisibility Problems
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12 is divisible by 3. 24,700 is a multiple of 100. x/15 is an integer. 6 is a factor of 17k. All of this language — divisible, multiple, integer, factor — signals that you’re about to begin a divisibility problem. Do you find these problems intimidating? Do you sometimes have no idea where to start? If so, this article offers a simple, painless way of thinking about divisibility that you can use on a wide range of GRE problems.
Prime numbers are building blocks
Think of every whole number as being made of prime “building blocks”. The blocks are combined together by multiplying their values.
Every number is made of a single, unique set of building blocks. 2 x 2 x 5 is the only possible way to break 20 down into primes. Because 2 and 5 are prime, they can’t be broken down any further: in that sense, they’re like the atomic elements that make up a molecule.
In GRE terminology, you’d say that the prime factorization of 20 is 2 x 2 x 5. The prime factors of 20 are 2 and 5. (By the way, they have to make it completely clear in each problem whether they want you to include that second 2!)
Finding prime factors
Every divisibility problem is actually a problem about prime factors. The fundamental math skill involved in these problems is quickly breaking numbers down into their prime factors. To do this, use a prime tree.
Suppose you’d like to factor the integer 25,200. Choose any number that divides evenly into it. In this case, I’ve chosen 100, because it’s easy to work with. Don’t think that you need to use a small number — just go with whatever’s easiest to divide into 25,200. Use that value to break 25,200 into a product: 25,200 = 100 x 252.
Now, do the same thing to each of the smaller values.
At this point, one of the values is prime, and can’t be broken down any further. Keep going, until every number is broken down into primes:
The primes in the tree are the prime factors of 25,200:
Practice this skill until you can complete a prime factor tree in under 30 seconds. One good exercise is to simply find all of the prime factors of every number under 100. Time yourself, and see how quickly you can finish. If you can do it in under 8 minutes, you’re on point.
The golden rule of divisibility
This fundamental rule will help you work out any GRE divisibility problem:
a larger number is divisible by a smaller number
MEANS
the larger number contains all of the smaller number’s prime factors
For instance, 25,200 is divisible by 100. Look at the building blocks of those two numbers. All of 100’s building blocks are contained within 25,200.
However, 25,200 isn’t divisible by 250:
250 has a building block — the third 5 — that 25,200 doesn’t have. That means that if you tried to divide 25,200 by 250, you’d end up with a fraction, because not all of the primes would cancel out:
Applying the rule to a problem
The following problem is from the 5lb. Book of GRE Practice Problems:
If k is a multiple of 24 but not a multiple of 16, which of the following cannot be an integer?
(A) k/8
(B) k/9
(C) k/32
(D) k/36
(E) k/81
First, analyze the problem. If k is a multiple of 24, then k contains all of 24’s building blocks. If you did the exercise above, you already know that those building blocks are 2, 2, 2, and 3:
k might also contain other building blocks. Or, it might not. All you know thus far is that it contains three 2’s and a 3.
You also know that it isn’t a multiple of 16. That means that it doesn’t contain the building blocks of 16, which are 2, 2, 2, and 2. That is, there are already three 2’s in there, and there can’t be any more. If there were, then k would be divisible by 16.
In summary: k contains exactly three 2’s, at least one 3, and possibly some other primes you don’t know about. If that’s the case, what numbers could divide evenly into k? Check the answer choices, one by one.
(A) 8 divides evenly into k, since there are three 2’s in k. This will be an integer.
(B) 9 could divide evenly into k, since there could be two 3’s in k. This might be an integer.
(C) 32 can’t divide evenly into k. There are five 2’s in 32, but only three 2’s in k. k/32 definitely won’t be an integer, so (C) is the right answer.
What to do next
This article doesn’t tell you everything you need to know about divisibility on the GRE. For that, you’ll want the GRE Number Properties Strategy Guide! But here are a few major takeaways to use as you start practicing divisibility problems:
- Every number is made of prime ‘building blocks’. When you start a divisibility problem, split every large number into these building blocks.
- To find prime factors quickly, use a prime factor tree. Practice creating prime factor trees for a few minutes now, and save a ton of time when you do problems on test day.
- Every time you see a divisibility problem, remind yourself that ‘is divisible by’ just means ‘contains all the building blocks of’. Translate problems in this way, and they become much easier to think about.
Good luck – and if you have other tips for handling GRE divisibility problems, share them in the comments! 📝
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Chelsey Cooley
is a Manhattan Prep instructor based in Seattle, Washington. Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master’s degree in linguistics, a 790 on the GMAT, and a perfect 170/170 on the GRE.Check out Chelsey’s upcoming GRE prep offerings here.