Factors and Multiples - Complete Notes for JNV Entrance Test 2025

1. Introduction

In mathematics, factors and multiples are fundamental concepts that help us in understanding relationships between numbers. These concepts play an important role in simplifying arithmetic problems, solving word problems, and even in algebra. The JNV Entrance Test often includes questions on factors and multiples, so mastering these concepts is essential.

This section will cover the definitions, properties, and methods to find factors and multiples of numbers, along with practice questions to help you prepare effectively for the JNV Entrance Test 2025.

2. Factors

A factor of a number is any number that divides that number exactly, without leaving a remainder. In simpler terms, if you can divide a number N by a number X without leaving a remainder, then X is a factor of N.

How to Find Factors of a Number

To find the factors of a number, start by dividing the number by 1, then proceed to check all the integers up to the number itself. If a number divides the original number exactly (i.e., remainder is 0), then it is a factor.

For example:

Factors of 12: To find the factors of 12, check all numbers from 1 to 12.
$12 \div 1 = 12, \quad 12 \div 2 = 6, \quad 12 \div 3 = 4, \quad 12 \div 4 = 3, \quad 12 \div 6 = 2, \quad 12 \div 12 = 1$
Therefore, the factors of 12 are: 1, 2, 3, 4, 6, 12.

Properties of Factors

A factor always divides the number exactly with no remainder.
Every number has at least two factors: 1 and the number itself.
The factors of a number always come in pairs. For example, the factors of 12 can be paired as (1, 12), (2, 6), (3, 4).

Prime Factors

A prime factor is a factor of a number that is a prime number. For example, the prime factors of 12 are 2 and 3 because:

12 = 2 \times 2 \times 3

3. Multiples

A multiple of a number is the product of that number and any integer. In simpler terms, multiples of a number are the numbers you get when you multiply that number by 1, 2, 3, and so on.

How to Find Multiples of a Number

To find the first few multiples of a number, simply multiply the number by 1, 2, 3, 4, 5, ...

For example:

Multiples of 5: The first few multiples of 5 are:
$5 \times 1 = 5, \quad 5 \times 2 = 10, \quad 5 \times 3 = 15, \quad 5 \times 4 = 20, \quad 5 \times 5 = 25$
So, the first few multiples of 5 are 5, 10, 15, 20, 25, ....

Properties of Multiples

Every number has an infinite number of multiples.
The multiples of a number are always greater than or equal to the number itself.
The multiples of a number are spaced at regular intervals (the size of the number itself).

4. Relationship Between Factors and Multiples

The concepts of factors and multiples are closely related. A factor of a number divides that number evenly, whereas a multiple of a number is obtained by multiplying the number by another integer.

If X is a factor of Y, then Y is a multiple of X.
If X divides Y exactly, then X is a factor of Y, and Y is a multiple of X.

For example:

Factors of 12: 1, 2, 3, 4, 6, 12. These are the numbers that divide 12 exactly.
Multiples of 12: 12, 24, 36, 48, 60, .... These are the numbers that can be written as 12 × 1, 12 × 2, 12 × 3, ...

5. LCM (Least Common Multiple)

The LCM of two or more numbers is the smallest multiple that is common to all of them. It is useful when we need to find a common denominator or when we want to solve problems involving multiple events happening at different intervals.

How to Find the LCM

List the first few multiples of each number.
Find the smallest common multiple.

For example, to find the LCM of 4 and 6:

Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Multiples of 6: 6, 12, 18, 24, 30, ...
The LCM of 4 and 6 is 12.

Alternatively, you can use the formula:

\text{LCM} = \frac{\text{Product of the numbers}}{\text{HCF}}

where HCF is the Highest Common Factor.

6. HCF (Highest Common Factor)

The HCF of two or more numbers is the largest number that divides all of them exactly. It is also known as the GCD (Greatest Common Divisor).

How to Find the HCF

List the factors of each number.
Find the largest number that is a common factor.

For example, to find the HCF of 12 and 18:

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
The HCF of 12 and 18 is 6.

7. Practice Questions

Multiple Choice Questions (MCQs)

Which of the following is a factor of 24?
- a) 5
- b) 6
- c) 7
- d) 9
- Answer: b) 6
What is the LCM of 6 and 8?
- a) 48
- b) 24
- c) 36
- d) 18
- Answer: b) 24
Which of the following is a multiple of 4?
- a) 10
- b) 16
- c) 7
- d) 9
- Answer: b) 16
What is the HCF of 18 and 24?
- a) 12
- b) 6
- c) 8
- d) 18
- Answer: b) 6
The first four multiples of 7 are:
- a) 7, 14, 21, 28
- b) 7, 10, 15, 20
- c) 7, 14, 18, 21
- d) 7, 21, 28, 35
- Answer: a) 7, 14, 21, 28

True/False Questions

The product of the LCM and HCF of two numbers is equal to the product of the numbers. (True/False)
- Answer: True
The number 1 is a factor of every number. (True/False)
- Answer: True
The LCM of two numbers is always greater than or equal to their HCF. (True/False)
- Answer: True
The factors of 15 are 1, 3, 5, and 15. (True/False)
- Answer: True

Fill in the Blanks

The HCF of 36 and 60 is _______.
- Answer: 12
The LCM of 5 and 10 is _______.
- Answer: 10
The number 1 is a factor of every number. (True/False)
- Answer: True

Conclusion

Understanding factors, multiples, HCF, and LCM is crucial for the JNV Entrance Test 2025. These concepts are foundational in mathematics and are widely used in solving problems related to number theory, algebra, and word problems. Practicing questions on these topics will help you gain confidence and perform better in your exam.

By mastering these concepts, you can efficiently solve problems involving divisibility, fractions, and the relationships between different numbers.

Jawahar Navodaya Vidyalaya Samiti (JNVS) Entrance Exams