GRE Even and Odd Numbers: What it is, Properties and Practice Problems

GRE Even/Odd Numbers is a key topic in the GRE test, falling under the Integers section within GRE Arithmetic. An even number is an integer that is divisible by 2 without any remainder. Examples include -6, -4, -2, 0, 2, 4, 6, 8, and 10. An even number can be expressed in the form 2n, where n is an integer. On the other hand, odd numbers are not divisible by 2 and leave a remainder when divided by 2. Examples of odd numbers are -5, -3, -1, 1, 3, and 5. Odd numbers can be represented by the form 2n+1, where n is an integer. Even and odd numbers have unique properties, which we’ll explore in this article along with a variety of practice problems and solutions relevant to GRE Even and Odd Numbers.

Important Points to Remember  Zero is considered an even number. The rules for determining odd and even numbers apply to negative integers just as they do to positive ones. For example, integers like -1, -2, -3, -4, -5, and -6 follow the same odd/even properties as their positive integers. For multi-digit integers, whether a number is odd or even can be determined by checking the last digit: if the last digit is even, the number is even; if the last digit is odd, the number is odd. The sum of any two consecutive integers will always be odd.

Properties of GRE Even/Odd Numbers

GRE Even/Odd numbers have distinct properties that are crucial for solving problems in the GRE Quantitative section. Below, we outline the properties of even and odd numbers when performing basic arithmetic operations like addition, subtraction, multiplication, division and exponents. Memorizing these rules can help you calculate more quickly and efficiently.

Properties on Addition

Operation	Result	Examples
Even+Even	Even	4+2 = 6 (Even)
Even+Odd or Odd+Even	Odd	6+3 = 9 or 3+6 = 9 (Odd)
Odd+Odd	Even	3+3 = 6 (Even)

Properties on subtraction

Operation	Result	Examples
Even-Even	Even	4-2 = 2 (Even)
Even-Odd or Odd-Even	Odd	4-3 = 1, 3-4 = -1 (Odd)
Odd-Odd	Even	5-3 = 2 (Even)

Properties on Multiplication

Operation	Result	Examples
Even*Even	Even	6*6 = 12 (Even)
Even*Odd or Odd*Even	Even	6*3 = 18 or 3*6 = 18 (Even)
Odd*Odd	Odd	9*9 = 81 (Odd)

Properties on Division

Operation	Result	Examples
Even/Even	Can be even, odd or a fraction	12/6 = 6 (Even), 6/2 = 3(Odd) or 6/12 = ½
Even/Odd	Can be even or a fraction	6/3 = 2 (Even), 4/3 = 1.33,
Odd/Even	Can be a fraction	3/2 = 1.5 , 35/2 = 17 ½
Odd/Odd	Odd or fraction	9/3 = 3 (Odd), 7/5 = 1 ⅖

Properties on Exponents

• A positive even number raised to any power remains even.
• A positive odd number raised to any power remains odd.

See the below table to know other rules with a negative base.

Base	Exponent	Result	Example
Positive Even	Even or Odd	Even	23 is 2*2*2 = 8 (Even) 43 is 4*4*4 = 64 (Even)
Positive Odd	Even or Odd	Odd	32 is 3*3 = 9 (Odd) 53 is 5*5*5 = 125 (Odd)
Negative Even	Even	Positive Even	-42 is -4*-4 = 16 (Positive Even)
Negative Odd	Even	Positive Odd	-34 is -3*-3*-3*-3=81 (Positive Odd)
Negative Even	Odd	Negative Even	-23 is -2*-2*-2 = -8 (Negative Even)
Negative Odd	Odd	Negative Odd	-53 is -5*-5*-5 = -125 (Negative Odd)
Even/Odd whether it is positive or negative integer	Zero	Is always 1.	50 or -51 = 1.

Practice Questions on GRE Even and Odd Numbers

As mentioned earlier, questions on even and odd numbers appear within the GRE Integers section of the exam. GRE questions on even/odd numbers can come in three formats: Quantitative Comparison Questions, Multiple-Choice Questions, and Numeric Entry Questions. Below, we provide practice problems for each question type to help you prepare and understand these concepts better.

Quantitative Comparison Questions

Question 1

If x and y are odd integers, compare the following:

Quantity A: x+y
Quantity B: An even integer

Solution:

Since the sum of two odd integers is always even, x+y will be even.

Answer:  The two quantities are equal

Question 2

If p is an even integer, compare the following:

Quantity A: p3 
Quantity B: p-1

Possible Answers:

1) The relationship between the quantities cannot be determined.
2) Quantity A is larger.
3) The two quantities are equal.
4) Quantity B is larger.

Solution:

Raising an even integer to any power results in an even number, so p3  is even. Regarding quantity B, subtracting 1 from an even integer is odd.

Answer: Hence, quantity A is larger.

Question 3:

If x is an odd integer, compare the following:

Quantity A: x²
Quantity B: x+1

Solution:
The square of an odd integer is always odd. x+1 would be even since it is one more than an odd integer. Thus, without knowing the specific value of x, we can’t determine the relationship, as sometimes x2 could be greater, and sometimes x+1 could be greater.

Answer: The relationship cannot be determined from the information given

Multiple Choice Questions 

Question 4

If x and y are odd integers, which of the following must be true?

A) x+y is odd
B) x*y is even
C) x+y is even

Solution:

Since odd + odd = even and odd × odd = odd, only C) is correct.

Question 5

If n is an even integer and m is an odd integer, which of the following is always true?

A) n×m is odd
B) n×m is even
C) n+m is even

Solution:
Since even × odd = even, B) is correct.

Numeric Entry Questions

Question 6

If a and b are consecutive even integers, and a=12, what is the value of a+b?

Solution:

Since a and b are consecutive even integers, and a=12, b must be the next even integer, so b=14.

a+b=12+14=26 .

Answer: 26

Question 7

Let p be an odd integer such that p2 is a three-digit number. If p=15, what is the value of p2?

Solution:

Sincep=15, calculate  p2 = 225

Answer: 225

Question 8

If m is an even integer and n is an odd integer, and m=8 and n=5, what is the value of 
m×n+m?

Solution:

m×n=8×5=40
m×n+m=40+8=48

Answer: 48

Question 9

If m is an odd integer and n is an even integer, and x=7 and y=6, what is the value of (x+y)×(x−y)

Solution:

x+y=7+6=13, x−y=7−6=1
(x+y)×(x−y)=13×1=13

Answer: 13

Question 10:

Solve for 17d = 225

Solution

17d = 225
d = 225/17

Answer is d = 15. And to note that an odd integer divided by an odd integer yields an odd integer.

Tips to ace the GRE Even/Odd Number Questions

1. Learn and memorize the basic properties of operations with even and odd numbers.
2. Before starting to find the answer, understand the question properly, as misinterpretation can lead to wrong answers. 
3. For quantitative comparison questions, try substituting small even and odd numbers into the equation to quickly find the answer.
4. Pay attention to terms like “integer,” “consecutive,” and “alternate” in questions, as understanding these can help solve sequence problems more effectively.
5. Use elimination method wherever necessary, as it is a quick way to reach the correct answer.

By understanding and effectively applying the properties of GRE even and odd numbers, you can approach these problems with confidence on the exam. To learn more about the GRE, its format, or available coaching options, simply fill out this demo form, and one of our representatives will reach out to assist you.