Course: GMAT

## Odd and Even – Number Properties Extended Concept (Part 1)

### Introduction to Odd and Even Numbers

Number properties play a significant role in various sections of the GMAT, including quantitative problem-solving and data sufficiency. One of the key number properties that candidates should be familiar with is the classification of integers into odd and even numbers.

#### A. Definition of Odd and Even Numbers

Even Numbers: An integer is called an even number if it is divisible by 2 without leaving a remainder. In other words, if an integer ‘n’ can be expressed as n = 2k, where k is also an integer, then ‘n’ is an even number.

Examples of even numbers: -4, -2, 0, 2, 4, 6, 8

Odd Numbers: An integer is called an odd number if it is not divisible by 2 or leaves a remainder of 1 when divided by 2. In other words, if an integer ‘n’ can be expressed as n = 2k + 1, where k is an integer, then ‘n’ is an odd number.

Examples of odd numbers: -3, -1, 1, 3, 5, 7, 9

#### B. Operations with Odd and Even Numbers

Understanding how odd and even numbers interact with basic arithmetic operations is crucial for solving GMAT problems.

Even + Even = Even

Odd + Odd = Even

Even + Odd = Odd

Even – Even = Even

Odd – Odd = Even

Even – Odd = Odd

Odd – Even = Odd

Multiplication:

Even × Even = Even

Odd × Odd = Odd

Even × Odd = Even

Division: A division with odd and even numbers can be more complex since it depends on the exact values of the numbers involved. The quotient may or may not be an integer, and even if it is an integer, it may not always have the same parity as the dividend and divisor.

#### C. Special Properties

Consecutive Integers: Consecutive integers are integers that follow each other in sequence, with a difference of 1 between each pair of numbers. In any sequence of consecutive integers, there will always be an even number of odd and even numbers.

Zero (0): Zero is considered an even number since it can be divided by 2 without leaving a remainder (0 = 2 × 0).

In the next part, we will delve deeper into the properties and rules of odd and even numbers and examine some GMAT problems that utilize these concepts.

Odd and Even – Number Properties Extended Concept (Part 2)

#### A. Squares of Odd and Even Numbers

Squares of Even Numbers: The square of an even number is always even. This can be proven using the definition of an even number: Let n be an even number, so n = 2k (where k is an integer). Then, n2 = (2k)2 = 4k2 = 2(2k2). Since 2k2 is an integer, n2 is an even number.

Squares of Odd Numbers: The square of an odd number is always odd. This can be proven using the definition of an odd number: Let n be an odd number, so n = 2k + 1 (where k is an integer). Then, n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. Since 2k2 + 2k is an integer, n2 is an odd number.

#### B. Prime Numbers and Odd/Even Numbers

Prime numbers are integers greater than 1 that have only two factors, 1 and themselves. Except for the number 2, all other prime numbers are odd numbers. This is because even numbers greater than 2 have at least three factors: 1, 2, and themselves.

#### C. Factorization and Odd/Even Numbers

The factorization of a number is the process of breaking it down into its prime factors. When dealing with odd and even numbers, it’s essential to know that:

The prime factorization of an even number will always include the prime number 2.

The prime factorization of an odd number will not have any even prime factors, including 2.

### D. Remainders and Odd/Even Numbers

When dividing integers, the remainder can be used to determine whether the dividend or divisor is odd or even:

The remainder will always be even when an even number is divided by an odd number.

The remainder will always be odd when an odd number is divided by an even number.

### III. Applying Odd and Even Number Properties to GMAT Problems

#### A. Problem-Solving Strategy

Identify whether the problem deals with odd and even numbers.

Apply the properties and rules of odd and even numbers to simplify the problem.

Use logic and reasoning to deduce the answer based on the simplified problem.

#### B. Data Sufficiency Strategy

Analyze the statements and determine whether they provide information about the odd or even nature of the numbers involved.

Apply the properties and rules of odd and even numbers to the statements.

Determine whether the given information is sufficient to answer the question.

In the following parts, we will explore example GMAT problems that require the application of odd and even number properties and learn how to solve them effectively.

## Odd and Even – Number Properties Extended Concept (Part 3)

### IV. Example GMAT Problems Involving Odd and Even Numbers

Data Sufficiency Examples

Example 1:

If x and y are integers, is x2 × y an odd integer?

(1) x is odd, and y is odd.

(2) x2 is odd.

Solution:

Statement (1): x is odd, and y is odd. We know that the product of two odd numbers is always odd. Therefore, x2 will be odd (since x is odd), and the product of x2 and y will also be odd. Statement (1) is sufficient to answer the question.

Statement (2): x2 is odd. If x2 is odd, then x must also be odd (as the square of an odd number is always odd). However, we need to get information about y. If y is odd, then the product x2 × y will be odd, but if y is even, the product will be even. Statement (2) needs to be more sufficient to answer the question.

Example 2:

If a, b, and c are consecutive integers and a < b < c, what is the value of b?

(1) a is an even number.

(2) c is an odd number.

Solution:

Statement (1): a is an even number. Since a is even, we know that b is odd and c is even, as they are consecutive integers. However, we cannot determine the exact value of b from this information alone. Statement (1) is not sufficient.

Statement (2): c is an odd number. If c is odd, then b must be even, and a must be odd, as they are consecutive integers. Again, we cannot determine the exact value of b from this information alone. Statement (2) is not sufficient.

Combining both statements, we still cannot determine the exact value of b. Both statements together are not sufficient.

Example 2:

If x and y are integers, is x + y even?

(1) x is an even number.

(2) y is an odd number.

Solution:

Statement (1): x is an even number. We need to know the parity of y to determine if x + y is even or odd. Statement (1) needs to be more.

Statement (2): y is an odd number. We need to know the parity of x to determine if x + y is even or odd. Statement (2) needs to be more.

Combining both statements, we know that x is even and y is odd. The sum of an even and an odd number is always odd. Both statements together are sufficient.

In the final part, we will discuss some tips for mastering odd and even number properties on the GMAT and provide additional practice problems for further understanding.

## Odd and Even – Number Properties Extended Concept (Part 4)

### Tips for Mastering Odd and Even Number Properties on the GMAT

Please familiarize yourself with the basic properties and rules of odd and even numbers and the results of their interactions with arithmetic operations. Be comfortable with the definitions and how to identify them and apply the properties in different situations.

Practice a variety of GMAT problems that involve odd and even numbers to gain a deeper understanding of their application in problem-solving and data sufficiency questions. This will help you identify patterns and develop a strong intuition for solving problems quickly and accurately.

Learn how to break down complex problems involving odd and even numbers into simpler components. Use the properties and rules to simplify problems and deduce the required information.

Remember that zero is considered an even number. This is a common source of confusion and can lead to mistakes in calculations and reasoning.

## Conclusion

Understanding the properties of odd and even numbers and their interactions with arithmetic operations is crucial for tackling GMAT problems effectively. By mastering these concepts and practicing various problems, you can significantly improve your quantitative reasoning skills and increase your chances of success on the GMAT.

## Practice questions with an answer key.

Practice Question 1:

If x and y are integers, what is the parity of x × y?

(A) Odd, if x and y are both odd

(B) Even, if x and y are both odd

(C) Even if x and y are both even

(D) Odd if x and y have different parity

(E) Even, if x and y have different parity

Practice Question 2:

What is the result of (-3) × (-4) × 5?

(A) -60 (B) -20 (C) 20 (D) 60 (E) 120

Practice Question 3:

Which of the following numbers must be divisible by 2?

(A) The sum of two odd numbers

(B) The difference between two odd numbers

(C) The product of two odd numbers

(D) The square of an odd number

(E) The sum of an odd number and an even number

Practice Question 4:

If a and b are consecutive even integers, what is the remainder when their product is divided by 4?

(A) 0 (B) 1 (C) 2 (D) 3 (E) Cannot be determined

Practice Question 5:

Which of the following is NOT true about prime numbers?

(A) All prime numbers are greater than 1

(B) All prime numbers are odd numbers

(C) All prime numbers have exactly two factors

(D) The number 2 is the smallest prime number

(E) The number 3 is a prime number

Practice Question 6:

If x, y, and z are consecutive odd integers and x < y < z, what is the remainder when their product is divided by 8?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

Practice Question 7:

If a and b are integers and a3 × b2 is an even number, which of the following must be true?

(A) a is even, and b is even

(B) a is even or b is even

(C) a is odd and b is odd

(D) a is odd or b is odd

(E) a is odd, and b is even

Practice Question 8:

Which of the following numbers is NOT divisible by both 2 and 3?

(A) 12 (B) 18 (C) 24 (D) 30 (E) 32

Practice Question 9:

If a, b, and c are consecutive integers and a < b < c, what is the remainder when the product abc is divided by 6?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

Practice Question 10:

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

(A) x, y and z are all odd

(B) x, y, and z are all even

(C) Exactly one of x, y, and z is odd

(D) Exactly one of x, y, and z is even

(E) Exactly two of x, y, and z are odd

Practice Question 11:

If a, b, and c are integers and a3 × b4 × c5 is an odd number, which of the following must be true?

(A) a, b, and c are all even

(B) a, b, and c are all odd

(C) a is odd, and b and c are even

(D) a and b are odd, and c is even

(E) a and b are even, and c is odd

Practice Question 12:

If x and y are integers, and x2 + y2 is an odd number, which of the following must be true?

(A) x is odd, and y is odd

(B) x is even, and y is even

(C) x is odd, and y is even

(D) x is even, and y is odd

(E) None of the above

Practice Question 13:

If a, b, and c are consecutive integers, and a < b < c, what is the remainder when the product a2 × b2 × c2 is divided by 8?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

Practice Question 15:

How many two-digit odd integers are divisible by the sum of their digits?

(A) 3 (B) 5 (C) 7 (D) 9 (E) 11

Practice Question 16:

If x and y are integers, and the expression x2 × y3 is divisible by 8, which of the following CANNOT be a value of y?

(A) -4 (B) 0 (C) 2 (D) 4 (E) 6

Practice Question 17:

For how many integers x between 1 and 100, inclusive, is 5x2 – 6x + 1 an even number?

(A) 25 (B) 50 (C) 75 (D) 90 (E) 100

Practice Question 18:

If x, y, and z are integers, and x3 × y4 × z5 is an even number, which of the following must be true?

I. At least one of x, y, and z is even

II. x3 × y4 × z5 is divisible by 8

III. x3 × y4 × z5 is divisible by 16

(A) I only

(B) I and II only

(C) I and III only

(D) II and III only

(E) I, II, and III

Practice Question 20:

If a, b, and c are distinct integers such that a2 + b2 = c2, what is the parity of ABC?

(A) abc is always even

(B) abc is always odd

(C) abc is even if a, b, and c are all even

(D) abc is odd if a, b, and c are all odd

(E) abc is even if exactly one of a, b, or c is odd

Practice Question 21:

If a, b, and c are consecutive integers and a < b < c, what is the remainder when the sum (a^3 + b^3 + c^3) is divided by 9?

(A) 0 (B) 1 (C) 2 (D) 3 (E) Cannot be determined

## Data Sufficiency (Five answer choices; one correct answer) Difficulty level: Easy

Practice Question 1:

Is x an even integer?

(1) x + 1 is odd

(2) x – 1 is odd

Practice Question 2:

Is the product of integers a, b, and c even?

(1) a is even

(2) b and c are both odd

Practice Question 3:

If x and y are integers, is x × y an odd number?

(1) x is odd

(2) y is odd

Practice Question 4:

Is the sum of three consecutive integers divisible by 3?

(1) The first integer is even

(2) The third integer is odd

Practice Question 5:

Is the sum of integers x and y even?

(1) x and y have the same parity

(2) x is divisible by 2

Practice Question 6:

If x, y, and z are integers, is the product XYZ odd?

(1) The sum x + y + z is odd

(2) Two of x, y, and z are even

Practice Question 7:

If a and b are consecutive even integers, what is the remainder when their product is divided by 8?

(1) a is divisible by 4

(2) b is divisible by 4

Practice Question 8:

If x and y are integers, is the sum x + y even?

(1) x2 is even

(2) y2 is odd

Practice Question 9:

If x and y are integers, is x3 × y2 an even integer?

(1) x is odd

(2) y is even

Practice Question 10:

If a, b, and c are integers, is the product (a2 × b × c3) an odd integer?

(1) a and c are odd

(2) b is even

Practice Question 11:

If x, y, and z are integers, is x3 × y2 × z5 an even number?

(1) x3 × z5 is odd

(2) y2 is even

Practice Question 12:

If a, b, and c are integers, is the expression (a2 × b3 × c4) an even integer?

(1) a is even

(2) b + c is odd

Practice Question 13:

If x and y are integers, is the expression x2 × y3 an odd integer?

(1) x2 is odd

(2) y3 is odd

Practice Question 14:

If x, y, and z are integers, and x < y < z, is the product XYZ divisible by 6?

(1) x, y, and z are consecutive integers

(2) The sum x + y + z is divisible by 3

Practice Question 15:

If a, b, and c are integers, is the sum a2 + b2 + c2 an even number?

(1) a + b + c is odd

(2) ab + ac + bc is even

Practice Question 16:

If x, y, and z are integers, is the product (x3 × y4 × z5) an even number?

(1) x and z are odd integers

(2) The product (x × y × z) is even

Practice Question 17:

If x and y are integers, is x2 × y3 an even number?

(1) x is odd and y is even

(2) x + y is odd

Practice Question 18:

If a, b, and c are integers, is the expression (a2 × b3 × c4) an even integer?

(1) b3 is odd

(2) a2 + c4 is even

Practice Question 19:

If x, y, and z are integers, is the expression x3 × y4 × z5 an odd integer?

(1) x and y are odd

(2) x3 × y4 is odd