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How the MAT Tests Math, Part 2: Primes and Factors

In “How the MAT Tests Math, Part 1,” I laid the groundwork for understanding how the MAT does (and doesn’t) test math. This time, we’ll take a closer look at some of the math topics you’re most likely to see. Some of these will likely be familiar, but don’t be fooled. Because they have to fit into so little space, MAT math questions often feature mathematical properties that are easy to state, but harder to test.

Primality, the quality of being a prime number, is a great example of such a property. So is the related notion of divisibility by a particular number. In an MAT math analogy, the relationships between the terms might be as simple yet obscure as a shared factor (“17 and 51 are both multiples of 17, just as 38 and 76 are multiples of 19.”)

Prime Time for an Example

To see how these deceptively simple properties creep into MAT problems, consider the following:

64 : 67 :: 128 : _________

(A) 121
(B) 124
(C) 125
(D) 127

When approaching a problem like this – numbers in every slot – it’s good to take a moment and brainstorm some possible rules that might relate the numbers:

Add 3. 64 + 3 = 67.
Multiply by 2. 64 × 2 = 128.

But you’ll quickly find that there isn’t a basic arithmetical operation that does the job. If we apply the rule “Add 3” to 128, we get 131, which isn’t on our list. If we apply the rule “Multiply by 2” to 67, we get 134 – again, not on our list. Other, more farfetched attempts to apply addition and multiplication just give us the same results. And it’s hard to see how exponentiation would help us here, since 67 is clearly not an integer power of 64.

So what’s going on? Typically of MAT math (and, to be honest, of the MAT in general), the answer might seem to have been “hidden in plain sight” once it’s revealed. 64 and 128 are composite, the opposite of prime. Each has positive integer factors other than itself and 1. (Actually, both 64 and 128 are powers of 2, but we don’t need to know that to solve the problem.)

67, on the other hand, is prime. So if there’s one and only one prime on the list, then that’s our answer! If there are multiple prime numbers on the answer list, then it’s back to the drawing board.

Let’s check:

121 is composite, because 11 × 11 = 121.
124 is composite, because 2 × 62 = 124.
125 is composite, because 5 × 25 = 125.
But 127 is prime. It has no positive integer factors other than itself and 1.

So there’s our analogy: “64 and 128 are composite, just as 67 and 127 are prime.”

More Prime Material for MAT Math

Remember that in math, as in every other content area, the MAT tends to favor knowledge of terms and concepts over knowledge of procedures. To that end, a question about primality might test to see if you “get” the underlying idea, rather than how well you can apply it. That’s what’s going on in this question:

prime : composite :: atom : _________

(A) neutron
(B) molecule
(C) particle
(D) element

In this math-chemistry hybrid, what counts is that we understand the relationship between prime and composite numbers. We might capture that relationship as follows:

Primes can be multiplied together to create composites, but not vice versa.
Primes are the building blocks that make up composite numbers.

Now, we just have to grab that concept and carry it to the other side of the analogy:

Atoms can be combined to create _________, but not vice versa.
Atoms are the building blocks that make up _________.

At this point, you may be thinking that molecule (B) seems like the best answer. And you’d be correct! After all, molecules are made up of atoms. Neutrons are a subatomic particle, so they don’t fit into the analogy. The word particle is hopelessly vague, including objects that are much smaller than atoms (e.g. quarks) as well as objects that are much larger.

“But wait,” I can hear some of you saying, “couldn’t you say that elements are made up of atoms?” In a sense, that’s true, but the analogy to molecules is more robust: a composite number, like a molecule, can be made up of different primes (atoms) or multiple copies of the same prime (atom). A number like 8, whose prime factorization is 23, would be analogous to a molecule of ozone (O3), which contains only oxygen atoms. A number like 12 (3 × 22) would be analogous to carbon dioxide (CO2), which contains both carbon and oxygen. Molecule is our best choice because it results in the stronger analogy.


We’re not done with our exploration of MAT math. In fact, we’re just beginning! The MAT will ask you a little bit of everything, so that’s what I plan to cover. In the weeks ahead, I’ll be posting about algebra, geometry, and basic statistics, with plenty of MAT-flavored problems to illustrate. As always, I recommend our free GRE Math Formula eBook if you need a quick refresher on any of the concepts in play here.

Until next time! 🙂

About Michael

A fan of logic games, classic RPGs, and espresso (in no particular order), Michael likes to think of standardized tests as puzzles to be pulled apart and analyzed. He delights in showing students the trapdoors, pitfalls, and shortcuts that lurk in even the simplest of questions. When he isn't helping students navigate the GRE, GMAT, or MAT, Michael enjoys writing poetry, tinkering with microcontrollers, and taking long, leisurely hikes in the woods.

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