What are the Most Frequently Asked GRE Number Theory Questions?

Ever wondered why GRE number theory questions often trip up even the most prepared GRE exam takers? These problems dive deep into integers, prime numbers, and divisibility rules, topics that many haven’t touched since high school.

The real challenge isn’t just the math itself, but the sneaky way these questions test your ability to think critically under pressure. Many students struggle, not because they don’t understand the concepts, but because they get lost in the details. Mostly, they fail to understand the benefits offered by GRE. The solution? A bit of focused practice and the right strategies can make these tricky problems much more manageable.

What is GRE Number Theory?

GRE number theory refers to questions in the GRE math section that focus on the properties and relationships of integers. Topics include concepts like prime numbers, divisibility, greatest common divisor (GCD), least common multiple (LCM), and remainders.

These questions test your understanding of basic number properties and often appear in both problem-solving and quantitative comparison formats. Understand the basic GRE math conventions to make things easier for you.

What to Expect From GRE Math Practice Questions?

When practicing GRE math questions, expect to solve a range of problems testing your grasp on quantitative reasoning. You’ll encounter tasks comparing two quantities, such as quantity A versus quantity B, requiring you to simplify expressions, apply formulas, and choose the correct answer from multiple answer choices.

Sample questions will often cover everything from basic arithmetic to algebraic equations like 2x = 120. GRE geometry and other such questions can also be encountered here.

The GRE quantitative section doesn’t just test knowledge, but how efficiently you can tackle problems without over-relying on the calculator. For example, a sample question might ask you to solve for the value of n in a given formula. With consistent practice, you’ll learn how to navigate the exam’s structure and improve your score across various question types. Also, students need to understand the basic difference between a GRE general and a subject test.

GRE Maths Questions (Sample Question) For Practice

When preparing for the GRE, practicing with sample math questions is one of the most effective ways to boost your performance. These questions give you a clear idea of what to expect on the test, from basic arithmetic, and GRE functions to more complex quantitative comparisons.

Below, you’ll find sample GRE math questions designed to challenge your understanding and improve your ability to solve problems under exam conditions. Let’s dive in and tackle these practice questions to sharpen your skills before test day.

Question 1: Word Problem – Pets Sold

A certain pet store sells only dogs and cats. In March, the store sold twice as many dogs as cats. In April, the store sold twice the number of dogs that it sold in March, and three times the number of cats that it sold in March. If the total number of pets the store sold in March and April combined was 500, how many dogs did the store sell in March?

Choices:
(A) 80
(B) 100
(C) 120
(D) 160
(E) 180

Solution:
Let the number of cats sold in March be cc, then the number of dogs sold in March is 2c2c.
In April, the store sold:

• Dogs: 2(2c)=4c2(2c)=4c
• Cats: 3c3c

Total pets sold in March and April:
2c+c+4c+3c=10c

Setting this equal to 500:
10c=500
c=50

Now, the number of dogs sold in March:
2c=2(50)=100

Answer: (B) 100

Question 2: Commission Calculation

Each month, Renaldo earns a commission of 10.5% of his total sales for the month, plus a salary of $2,500. If Renaldo earns $3,025 in a certain month, what were his total sales?

Choices:
(A) $80,000
(B) $100,000
(C) $120,000
(D) $160,000
(E) $180,000

Solution:
Let SS be the total sales. The total earnings can be expressed as:
0.105S+2500=3025

Subtracting $2,500 from both sides:
0.105S=3025−2500
0.105S=525

Now, divide both sides by 0.105:
S=525/0.105=5000

Answer: (B) $100,000

Question 3: Quantitative Comparison

The average (arithmetic mean) high temperature for x days is 70 degrees. The addition of one day with a high temperature of 75 degrees increases the average to 71 degrees.

Quantity A: x
Quantity B: 5

Choices:
(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.

Solution:
The total temperature for x days is:
70x

After adding one more day at 75 degrees, the total becomes:
70x+75

The new average is given by:
(70x+75) / (x+1) = 71

Cross-multiplying gives:
70x+75=71(x+1)
70x+75=71x+71

Rearranging gives:
75−71=71x−70x

4=x

Comparison:
Quantity A = 4
Quantity B = 5

Answer: (B) Quantity B is greater.

Question 4: Geometry – Circle

A circle has a radius of 10 units. What is the area of the circle?

Choices:
(A) 100π
(B) 200π
(C) 300π
(D) 400π
(E) 500π

Solution:
The area AA of a circle is given by the formula:
A=πr2

Substituting the radius:
A=π(102)=100π

Answer: (A) 100π

Question 5: Ratio Problem

If the ratio of the lengths of two sides of a rectangle is 3:5 and the perimeter is 64 units, what is the length of the longer side?

Choices:
(A) 20
(B) 24
(C) 30
(D) 32
(E) 36

Solution:
Let the lengths be 3x and 5x. The perimeter P of a rectangle is given by:

P=2(length+width)

64=2(3x+5x)
64=2(8x)
64=16x
x=4

Thus, the longer side is:
5x=5(4)=20

Answer: (A) 20

Question 6: Algebra – Quadratic Equation

What are the roots of the equation x²−5x+6=0?

Choices:
(A) 1 and 6
(B) 2 and 3
(C) -2 and -3
(D) 0 and 6
(E) 1 and 5

Solution:
Factoring the quadratic:
(x−2)(x−3)=0
Setting each factor to zero gives the roots:
x−2=0⇒x=2
x−3=0⇒x=3

Answer: (B) 2 and 3

Question 7: Percentage Problem

A shirt originally priced at $60 is on sale for 25% off. What is the sale price of the shirt?

Choices:
(A) $45
(B) $50
(C) $55
(D) $60
(E) $65

Solution:
The discount amount is:
0.25×60=150.25×60=15
The sale price is:
60−15=4560−15=45

Answer: (A) $45

Question 8: Sequence

What is the 10th term in the sequence 2, 5, 8, 11, …?

Choices:
(A) 20
(B) 23
(C) 26
(D) 29
(E) 32

Solution:
The sequence has a common difference of 3.
The n-th term of an arithmetic sequence is given by:
a_n=a₁+(n−1)⋅d
Where:

• a₁=2 (first term)
• d=3 (common difference)
• n=10

Substituting the values:
a₁₀=2+(10−1)⋅3
=2+9⋅3=2+27=29

Answer: (D) 29

Question 9: Mixture Problem

A chemist has two solutions, one with a concentration of 40% acid and the other with a concentration of 60% acid. How many milliliters of the 40% solution must be mixed with 30 milliliters of the 60% solution to produce a solution with a concentration of 50% acid?

Choices:
(A) 10 mL
(B) 15 mL
(C) 20 mL
(D) 25 mL
(E) 30 mL

Solution:
Let the amount of the 40% solution be xx mL.
The amount of acid in the 40% solution is 0.4×0.4x mL.
The amount of acid in the 60% solution is 0.6(30)=180.6(30)=18 mL.
The total amount of acid in the mixture is 0.4x+180.4x+18 mL.
The total amount of the mixture is x+30x+30 mL.

The concentration of the mixture is 50%, so:
(0.4x+18)/(x+30)=0.5

Solving for x:
0.4x+18=0.5x+15
0.1x=−3
x=−30

Since the amount of the 40% solution cannot be negative, we need to find the positive solution.

0.4x+18=0.5x+15
0.1x=−3
x=−30

Answer: (A) 10 mL

Question 10: Geometry – Triangles

In triangle ABC, AB=6 and BC=8. What is the length of AC?

Choices:
(A) 2
(B) 4
(C) 6
(D) 8
(E) 10

Solution:
In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

AB+BC>AC
6+8>AC
14>AC
AC<14

Since AB=6 and BC=8, the maximum possible value for AC is 14 – 2 = 12.

The minimum possible value for AC is the absolute value of the difference between AB and BC, which is ∣6−8∣=2∣

Therefore, the length of AC is between 2 and 12, inclusive.

Answer: (A) 2​

Question 11: Data Interpretation – Mean, Median, Mode

The mean of a set of 10 numbers is 20. If one of the numbers is replaced by 30, the mean becomes 22. What is the median of the original set of numbers?

Choices:
(A) 10
(B) 15
(C) 20
(D) 25
(E) 30

Solution:
Let the sum of the original set of numbers be SS. Then:
S/10=20
S=200

After replacing one number with 30, the new sum is S−x+30, where x is the replaced number. The new mean is 22, so:
(S−x+30)/(10)=22

S−x+30​=220
S−x=190
x=10

The median of the original set is the middle number when the set is arranged in ascending order. Since the mean is 20, the median is also 20.

Answer: (C) 20

Question 12: Algebra – Solving Equations

Solve for x in the equation:
3(x−2)+4=2(x+1)

Choices:
(A) x=1
(B) x=2
(C) x=3
(D) x=4
(E) x=5

Solution:

1. Distribute on both sides:
   3x−6+4=2x+2
   3x−2=2x+2
2. Subtract 2x from both sides:
   3x−2x−2=2
   x−2=2
3. Add 2 to both sides:
   x=4

Answer: (D) x=4 

5 Expert Tips To Excel At GRE Quantitative Section

The GRE Quant section often intimidates test-takers because it includes tough GRE math questions and multiple formats like quantitative comparison questions, numeric entry questions, and multiple-choice questions.

With the right GRE prep strategy, however, you can strengthen your skills and score higher in the GRE® exam, making you stand out for graduate schools.

1. Master the Format of GRE Quant Section

The GRE math section includes quantitative comparison questions, GRE multiple-choice questions, and GRE numeric entry questions. Knowing that quantitative comparison questions ask you to compare values like the value of x, or a positive integer, helps you save time during the exam. Practice with free GRE practice tests from ETS to see how questions are similar to the real test.

2. Build Strong Fundamentals in Numbers and Operations

Many difficult GRE questions require quick recall of basics like how to multiply fractions, take a square root, or simplify an exponent. You may be asked to find the LCM, calculate HCF, or deal with consecutive integers. Strengthening these skills through GRE quantitative practice ensures you can solve even a term of the sequence or satisfy the equation under timed pressure.

3. Learn to Handle Word Problems Effectively

The GRE often disguises concepts in real-life scenarios, like calculating the number of students, the diameter of the smaller circle, or the radius of the smaller figure. GRE quantitative comparison questions might involve finding if AB must equal a side length, or solving problems around percent, decimal, and 0.5 conversions. These require careful reading, so always break down what questions you’ll face step by step.

4. Practice With Increasing Levels of Difficulty

The GRE quant sections range in level of difficulty, starting with simpler problems and moving to hard GRE math practice questions. Let’s take an example: If asked to evaluate 216 or to look at 15 consecutive integers, the key is to simplify step by step. Using free GRE and self-study resources for GRE practice test prep helps you tackle the gamut of problems, from easy to advanced.

5. Use Smart Test Prep Strategies and Review Mistakes

In your test prep, use practice tests and track how many 15 questions you answer correctly in each set. Focus on multiple-answer types and numeric entry questions that’ll challenge your reasoning. Compare your GRE general practice performance to another GRE attempt to see improvement. Reviewing mistakes, especially in difficult GRE math questions, will strengthen your approach for when you must be chosen to attempt the toughest items.

Conclusion

Excelling at the GRE quantitative section requires focused preparation, logical problem-solving, and consistent practice with various question types. By incorporating the strategies we’ve outlined, building a strong foundation, using GRE coaching, tackling sample questions, and simulating test day conditions, you’ll be well-equipped to handle even the most difficult GRE math problems. Stay consistent, and with the right approach, cracking the GRE is entirely within reach.

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