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Introduction to Factors | The Complete SAT Course - Class 10 PDF Download

Introduction

Factors are an essential concept in mathematics and play a significant role in the SAT exam. A factor is a number that divides another number exactly without leaving a remainder. In other words, factors are the building blocks or divisors of a given number. A proper understanding of factors is crucial for solving various algebraic and arithmetic problems in the SAT exam. This document covers the essential concepts related to factors, including examples and practice questions with solutions to help you master this topic.

Basic Concepts and Definitions

1. Factors: Factors are the numbers that can be multiplied together to get another number, also known as the product.
For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because:

1 × 12 = 12
2 × 6 = 12
3 × 4 = 12

2. Prime Numbers: Prime numbers are numbers greater than 1 that have only two factors, namely 1 and the number itself.
For example, 

The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, etc.

3. Composite Numbers: Composite numbers are numbers greater than 1 that have more than two factors. In other words, they can be divided by at least one number other than 1 and themselves.
For example,

Some composite numbers are 4, 6, 8, 9, 10, etc.

4. Greatest Common Factor (GCF): The GCF of two numbers is the largest number that divides both numbers exactly.
For example, 

The GCF of 20 and 30 is 10.

5. Least Common Multiple (LCM): The LCM of two numbers is the smallest number that is a multiple of both numbers.
For example, 

The LCM of 4 and 6 is 12.

Examples and Explanations

1. Finding Factors of a Number:
Q. Find the factors of 24.

To find the factors of 24, we can consider the pairs of numbers whose product is equal to 24.
1 × 24 = 24
2 × 12 = 24
3 × 8 = 24
4 × 6 = 24
So, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

2. Finding the Greatest Common Factor (GCF):
Q. Find the GCF of 36 and 48.

First, find the factors of both numbers.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Now, find the common factors of both numbers.
Common factors: 1, 2, 3, 4, 6, 12
The largest common factor is 12. Therefore, the GCF of 36 and 48 is 12.

3. Finding the Least Common Multiple (LCM):
Q. Find the LCM of 5 and 6.

First, list the multiples of both numbers.
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, …
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, …
The smallest common multiple is 30. Therefore, the LCM of 5 and 6 is 30.

Practice Questions with Solutions

Q.1. Find the factors of 18.

1 × 18 = 18
2 × 9 = 18
3 × 6 = 18
So, the factors of 18 are 1, 2, 3, 6, 9, and 18.

Q.2. Find the GCF of 21 and 14.

Factors of 21: 1, 3, 7, 21
Factors of 14: 1, 2, 7, 14
Common factors: 1, 7
The GCF of 21 and 14 is 7.

Q.3. Find the LCM of 8 and 12.

Multiples of 8: 8, 16, 24, 32, 40, 48, …
Multiples of 12: 12, 24, 36, 48, 60, …
The smallest common multiple is 24. Therefore, the LCM of 8 and 12 is 24.

Conclusion

The concept of factors is fundamental to understanding many mathematical problems in the SAT exam. By mastering the basics of factors, prime numbers, composite numbers, GCF, and LCM, you will be well-equipped to tackle a variety of questions in the Maths section of the SAT exam. Practice and familiarize yourself with these concepts to improve your problem-solving skills and score high on the SAT.

The document Introduction to Factors | The Complete SAT Course - Class 10 is a part of the Class 10 Course The Complete SAT Course.
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