Notes on Even Numbers and Odd Numbers | JNVS 2025

 

Notes on Even Numbers and Odd Numbers

1. Even Numbers

An even number is any integer that is exactly divisible by 2. In other words, when an even number is divided by 2, there is no remainder. Mathematically, an even number can be expressed as:
$\text{Even number} = 2n$
Where n is an integer (positive, negative, or zero).

Characteristics of Even Numbers

• The last digit of any even number will always be 0, 2, 4, 6, or 8.

• Every second number in the number line is an even number. For example, 2, 4, 6, 8, 10, 12, 14, ....

• 0 is also considered an even number, since it is divisible by 2 (i.e., 0 ÷ 2 = 0).

• The sum of two even numbers is always even.

• The product of any number and an even number is always even.

Examples of Even Numbers

• 4, 12, 200, -8, 0, 22.

Even Numbers in Real Life

• Pairs of objects: When items come in pairs, such as shoes, socks, or chairs, the number of items is usually an even number.

• Time: Even numbers often represent even minutes on a clock (e.g., 2:00, 4:00, 6:00, etc.).

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2. Odd Numbers

An odd number is an integer that cannot be exactly divided by 2. In other words, when an odd number is divided by 2, there is always a remainder of 1. Mathematically, an odd number can be expressed as:
$\text{Odd number} = 2n + 1$
Where n is an integer.

Characteristics of Odd Numbers

• The last digit of any odd number will always be 1, 3, 5, 7, or 9.

• Odd numbers appear at alternate positions in the number line. For example, 1, 3, 5, 7, 9, 11, 13, ....

• The sum of two odd numbers is always even.

• The product of an odd number and an even number is always even, but the product of two odd numbers is always odd.

Examples of Odd Numbers

• 1, 3, 5, 13, -9, 15.

Odd Numbers in Real Life

• Unpaired objects: When items are left unpaired, such as a single shoe, sock, or apple, the number of items is usually odd.

• Time: Odd numbers often represent odd minutes on a clock (e.g., 1:00, 3:00, 5:00, etc.).

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3. Key Differences Between Even and Odd Numbers

Property	Even Numbers	Odd Numbers
Divisibility	Divisible by 2	Not divisible by 2
Last Digit	0, 2, 4, 6, 8	1, 3, 5, 7, 9
Sum	Sum of two even numbers is even	Sum of two odd numbers is even
Product	Product of two even numbers is even	Product of two odd numbers is odd
Examples	2, 4, 6, 8, 10, 12	1, 3, 5, 7, 9, 11

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Practice Questions on Even and Odd Numbers

Multiple Choice Questions (MCQs)

1. Which of the following is an even number?

   • a) 15

   • b) 22

   • c) 37

   • d) 51

   • Answer: b) 22

2. Which of the following is an odd number?

   • a) 4

   • b) 8

   • c) 13

   • d) 28

   • Answer: c) 13

3. The sum of two odd numbers is:

   • a) Always odd

   • b) Always even

   • c) Sometimes odd, sometimes even

   • d) Cannot be determined

   • Answer: b) Always even

4. Which of the following cannot be an even number?

   • a) 0

   • b) 2

   • c) 13

   • d) 14

   • Answer: c) 13

5. Which of the following is an even number?

   • a) -4

   • b) -5

   • c) -7

   • d) -9

   • Answer: a) -4

6. The product of an even number and an odd number is:

   • a) Always even

   • b) Always odd

   • c) Sometimes even, sometimes odd

   • d) None of the above

   • Answer: a) Always even

7. What is the next odd number after 23?

   • a) 24

   • b) 25

   • c) 26

   • d) 27

   • Answer: b) 25

8. Which of the following is not an odd number?

   • a) 1

   • b) 5

   • c) 7

   • d) 14

   • Answer: d) 14

9. What is the sum of the first four even numbers?

   • a) 10

   • b) 12

   • c) 14

   • d) 16

   • Answer: b) 12

10. The difference between two consecutive odd numbers is always:

    • a) 1

    • b) 2

    • c) 3

    • d) 4

    • Answer: a) 1

True/False Questions

1. The number 0 is an odd number. (True/False)

   • Answer: False

2. The product of two even numbers is always odd. (True/False)

   • Answer: False

3. The sum of two even numbers is always even. (True/False)

   • Answer: True

4. An odd number can never be divisible by 2. (True/False)

   • Answer: True

5. Every even number is divisible by 2. (True/False)

   • Answer: True

Fill in the Blanks

1. The number _ is the smallest odd number.

   • Answer: 1

2. The product of two odd numbers is always _.

   • Answer: odd

3. The sum of an even and an odd number is always _.

   • Answer: odd

4. The last digit of an even number is always _.

   • Answer: 0, 2, 4, 6, or 8

5. _ is an example of an even number.

   • Answer: 4

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Conclusion

Understanding even and odd numbers is essential for solving a variety of problems in mathematics. By recognizing the properties and differences between them, you can easily solve questions related to basic arithmetic and number theory. The practice questions provided will help you test your understanding and improve your ability to solve these types of problems efficiently.

By practicing regularly and applying these concepts in different contexts, you will improve your skills and perform better in exams, including the JNV Entrance Test.