Number Properties on the GMAT
How long has it been since you learned about odd and even numbers? What about the differences between positives and negatives? And when did you learn the definition of an integer or a prime number?
Most of us learned all of these things before we were 10 years old. But as you may have noticed, the GMAT tests these concepts—collectively known as number properties—in ways that make them seem absolutely foreign. Here is some example question stems drawn from Data Sufficiency problems:
- Is x even?
- Is x < 0?
- Is x an integer?
In Problem Solving, algebra problems are frequently set up such that you will answer incorrectly if you assume x must be an integer or that it must be positive.
Number properties are all about categories and rules; certain kinds of numbers behave the same way in all cases. The GMAT will reward you for using the Core Competencies of Pattern Recognition and Critical Thinking to draw inferences about how numbers behave, based on certain characteristics or “properties” they possess. That’s why number properties questions appear on the Quant section with greater frequency than other topics.
What Types of Problems Fall Under GMAT Number Properties?
On the GMAT, Number Properties questions cover four broad categories. The first of those is divisibility and primes. Here are the key skills you’ll need:
- Know the language. Be able to define the following terms: prime, factor, multiple, integer, prime factor, and divisor.
- Know how and when to divide a number into its prime factors.
- Understand how a number’s prime factors relate to its divisibility.
The next topic is remainders. Remainders aren’t tested nearly as often as divisibility, although the two ideas are related! You should know:
- How to calculate remainders, and how to find numbers that have a particular remainder
- The clues that tell you to start thinking about remainders: words and phrases like ‘remaining,’ ‘left over,’ and ‘divided into groups’
GMAT Number Properties also covers odds and evens, and positives and negatives. These two topics go hand in hand. You’ll need to know what happens when you add, subtract, multiply, and divide odd and even, or positive and negative, numbers.
Finally, combinatorics and probability are rare and challenging GMAT Number Properties problem types.
Here are some definitions to brush up on and some tips for mastering the number properties concepts tested most frequently on the GMAT.
Integers
The term integer refers to positive whole numbers, negative whole numbers, and zero. When an integer is added to, subtracted from, or multiplied by another integer, the result is always an integer. (An integer divided by an integer may or may not result in an integer; it depends on whether the first number is a multiple of the second.) Picking Numbers makes questions about integers and non-integers easier to tackle.
Zero is a special case. It is an integer, and it is even, but it is neither positive nor negative. Zero has no sign. One is also somewhat special; 0 and 1 behave differently from all other integers when you multiply them, with one important exception: odd and even rules. Zero behaves the same as all other even integers when adding, subtracting, or multiplying, and 1 behaves the same as all other odds in these situations.
When “integer” is a central word in a question, you know have a number properties question. Questions that focus on the rules governing integers force test-takers to discriminate between different categories of numbers (whole numbers versus fractions or decimals). These questions also contain an important trap that you must learn to avoid: Never assume a number is an integer unless you’re told that it is. The absence of information in a GMAT question can be just as important as its inclusion.
Odd and Even
The terms odd and even apply only to integers. Even numbers are integers that are divisible by 2, and odd numbers are integers that are not. Odd and even numbers may be negative; 0 is even. The product of an even number and any integer will always be even. There are no universal rules for dividing odds and evens.
Positive and Negative
Positive numbers are greater than zero, falling to the right of 0 on a number line. Negative numbers are less than zero, falling to the left on a number line.
When multiplying or dividing two numbers that have the same “sign” (positive or negative), the result is always positive. When multiplying or dividing two numbers with different signs, the result is always negative.
Some GMAT questions hinge on whether the numbers involved are positive or negative. These properties are especially important to keep in mind when Picking Numbers on a Data Sufficiency question. If both positives and negatives are permissible for a given question, make sure you test both possibilities, since doing so will often yield significant (different) results. Take the same approach that you’ve been learning to use for other number properties: Spend some time memorizing the rules, but always keep your eye out for strategic opportunities to pick numbers.
The special properties of −1, 0, and 1 make them important numbers to consider in Data Sufficiency questions, as well as for the “could be/must be” kinds of Problem Solving questions. Because numbers between −1 and 1 behave differently than do other numbers, they are good numbers to pick when testing whether one expression always has to be less than or greater than another.
Divisibility Rules
Because the GMAT tests math concepts most of us learned in elementary school, it can be challenging to recall some of these basic quantitative building blocks. Boost your ability to do mental math with these tips for remembering how to divide numbers in your head.
Rules for Twos, Fives, and Tens
The easiest of the divisibility rules are the rules for 2, 5, and 10:
- If a number is even, it is divisible by 2.
- If a number ends in 0 or 5, it is divisible by 5.
- If a number ends in 0, it is divisible by 10.
Rules for Threes, Sixes, and Nines
If the sum of a number’s digits is divisible by 3, the number is divisible by 3. For example, 432 → 4+3+2=9, so 432 is divisible by 3. But 253 → 2+5+3=10, so 253 is not divisible by 3.
If a number is even AND the sum of its digits is divisible by 3, it is divisible by 6: 432 → 4+3+2=9, and 432 is even (divisible by 2), so it is divisible by 6.
If the sum of a number’s digits is divisible by 9, the number is divisible by 9: 432 → 4+3+2=9, so 432 is divisible by 9; 837 → 8+3+7=18, so 837 is divisible by 9.
An interesting side note about 9: All multiples of 9, when their digits are summed, eventually yield 9. For example, 837 → 8+3+7=18, and 1+8=9.
Rules for Fours
If the last two digits of a number are a multiple of 4, the entire number is a multiple of 4. You don’t add the digits together; if the last two are a multiple of 4 that you recognize, you can trust you have a multiple of 4. Because 100 is divisible by 4, all that matters is the last two digits. For example, 2,348,632 is divisible by 4 because the last two digits, 32, is divisible by 4.
I’m not aware of any useful ways to determine divisibility by 7 or 8, so your best bet for those is to learn their multiples instead. Most of us have forgotten our multiplication tables, so there is no shame in needing to refresh them. Take care to learn the multiples of 13 as well. Because most people brush up on multiples through 12, the GMAT loves to throw in 13 to catch you off guard.
Spend some time reviewing number properties. Being aware of and comfortable with the behaviors of numbers will take you a long way toward landing your best GMAT score on Test Day.
How to Review GMAT Number Properties Questions
The secret to reviewing a GMAT Number Properties problem is to break it into little pieces. It’s not (usually) the math that’s tough. The challenge is figuring out what the GMAT is trying to tell you, and doing so quickly and calmly.
You might not understand every piece of a problem the first time you see it, especially if you’re under pressure from using a timer. That’s what the review process is for. Try to spot the little clues and pieces of information in each problem and analyze what each one means. Your goal is to find any pieces of this problem that you might be able to use to solve other, different problems in the future.
As always, do as much of the hard work yourself as possible. That means starting by reviewing your own work, without looking at the answers. Then, just check the answer to see if you got it right. If not, look at the problem one more time. Now that you know the answer, can you figure it out?
GMAT Number Properties Practice Questions
These problems each test Number Properties topics, and each one contains some “GMAT code” that you’ll need to translate. Go ahead and work through them now! If you’re feeling brave, set a countdown timer for 12 minutes. That’s about how much time you’d have for these on test day.
GMAT Number Properties: Divisibility, Primes, and Remainders
1. Is x/10 an integer?
(1) x/40 is an integer
(2) x/5 is an integer
2. If 1000 is divisible by 5jk, j and k are positive integers, and j > k, what is the largest possible value of k?
(A) 5
(B) 8
(C) 10
(D) 20
(E) 50
3. Is the number of students in a certain club divisible by 15?
(1) If the club were divided as evenly as possible into six teams, three of the teams would each have one extra student.
(2) The club can be evenly divided into teams of five students each, with no students left over.
GMAT Number Properties: Evens, Odds, Positives, and Negatives
4. If x and y are positive integers, is xy+1 even?
(1) x + y is even
(2) 3x is odd
5. If x < 0 < y, which of the following must be negative?
(A) (xy)²
(B) x² y
(C) x + y
(D) x² + xy
(E) xy²
6. If abc ≠ 0, is ab > 0?
(1) ab²c > 0
(2) abc² > 0
Practice Questions Answer Explanations
Problem 1:
This is a Data Sufficiency problem. The question is really asking whether x is a multiple of 10.
The first statement tells you that x is a multiple of 40. So, x could be a number like 40, 80, 400, or even 0 or -40. All of these numbers are multiples of 10. So, x is definitely a multiple of 10, and the first statement is sufficient.
However, the second statement only tells you that x is a multiple of 5. If x is a multiple of 5, it might be a multiple of 10, or it might not be. For instance, x could be 20, but it could also be 15. Since you don’t know whether x is a multiple of 10 or not, this statement is insufficient. The answer is (A).
Problem 2:
This is a difficult problem to untangle at first. However, what it’s really saying isn’t that complicated!
5jk divides evenly into 1000. So we can divide 1000 evenly by 5, j, and k.
We can go ahead and divide 1000 by 5, and we get 200. We don’t know what j and k are yet, but we must be able to divide 200 by them.
There are a lot of different pairs of numbers that you could divide 200 by. Of those pairs, we’re looking for the one that has the biggest value for k. Start writing out the possible pairs of divisors:
jk = 1*200
jk = 2*100
jk = 4*50
jk = 5*40
jk = 10*20
Since k has to be smaller than j, k must be the smaller number in the pair. Of all of these pairs, the one that has the greatest value for k is jk = 10*20, where k will equal 10.
Problem 3:
This problem is about remainders as well as divisibility. The question asks whether the number of students is divisible by 15. According to the first statement, if you divide the number of students by 6, you get a remainder of 3 (the number of leftover students.)
So the first statement says that the number of students could be 9, or 15, or 21, or any other number that has a remainder of 3 when divided into 6 groups. Some of these values are divisible by 15 and others aren’t. Since the number of students might or might not be divisible by 15, this statement is insufficient.
The second statement says that the number of students is divisible by 5. However, the number might be divisible by 15 (for instance, if it equals 15 or 30) or it might not be (if it equals 10 or 20). This statement is also insufficient.
Put the two statements together. Notice that all of the numbers that fit the first statement are multiples of 3. That isn’t a coincidence! Imagine dividing the students into three groups. You could create each group by combining two of the six smaller groups together, then adding one of the three leftover students. Since the students could be divided evenly into three groups, the number of students is a multiple of three.
Putting the two statements together tells us that the number of students is a multiple of both 3 and 5. If a number is a multiple of 3 and a multiple of 5, it’s a multiple of 15. The two statements are sufficient together and the answer is (C).
Problem 4:
When this problem asks whether xy + 1 is even, it’s really asking whether xy is odd. And since xy can only be odd if both x and y are odd, you can rewrite the question like this:
“Are x and y both odd?”
Statement 1 tells you that x and y are either both odd or both even. Since you don’t know whether they’re both odd, the statement is insufficient.
Statement 2 tells you that x is odd. However, y could be odd or even, so the statement is insufficient.
Putting the two statements together, x is odd (from statement 2), and x and y are the same (from statement 1). So, x and y are both odd, and xy + 1 is even. The two statements together are sufficient, and the answer is (C).
Problem 5:
The question says that x is negative and y is positive. Look through the answer choices to find one that will always turn out negative.
(A) can’t be negative, since perfect squares are never negative. Similarly, (B) is the product of two positive numbers, so (B) can’t be negative. Eliminate (A) and (B).
(C) could be either positive or negative. For instance, if x = -100 and y = 5, x + y is negative. But if x = -5 and y = 100, x + y is positive. (D) can also be positive—for instance, if x = -10 and y = 1. Eliminate (C) and (D).
(E) is the only answer choice that must be negative. y² is positive, and x is negative, so their product will be negative.
Problem 6:
The question states that abc does not equal 0, so none of the unknown values equals zero. The product of two numbers is positive if they’re both positive or both negative: in other words, if they have the same sign. So, you can rewrite the question:
“Do a and b have the same sign?”
Statement 1 states that ab²c > 0. b² is definitely positive, since it’s a square. Therefore, it’s safe to divide both sides of the inequality by b², which simplifies it to ac > 0. However, because there’s no information about the sign of a or the sign of b, this statement is insufficient.
Statement 2 states that abc² > 0. Since c² is definitely positive, you can divide both sides of the inequality by it, and find that ab > 0. This answers the original question! So, statement 2 is sufficient and the answer to the problem is (B).
