The best thing about GMAT Number Properties problems is that the numbers are nice and easy. There’s no need to worry about fractions, decimals, or percents!
On the flip side, whoever writes GMAT Number Properties questions loves to make simple concepts seem complicated. These problems are often written in “GMAT code” rather than plain English, and to start solving them, you need to translate them first.
If you do these tricky Number Properties practice problems, you’ll get to practice translating GMAT code, as well as working on some important math skills.
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. We won’t look at these problems here, because they’re covered in this article. They’re also rare and often difficult, which makes them great candidates for guessing!
If you need a refresher on the GMAT Number Properties rules and strategies, the Number Properties Strategy Guide is the first place to go. Foundations of GMAT Math also starts from scratch and covers the basics of divisibility.
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?
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?
(B) x² y
(C) x + y
(D) x² + xy
6. If abc ≠ 0, is ab > 0?
(1) ab²c > 0
(2) abc² > 0
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?
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).
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.
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).
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).
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.
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).
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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 GMAT prep offerings here.