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Factors of a Number

Factors are the fundamental units of numbers in mathematics. They assist us in decomposing big numbers into smaller, easier-to-manage components. Solving problems involving divisibility, simplification, LCM, HCF, algebra, and geometry requires an understanding of factors. Understanding the concept of factors is essential whether you're studying for competitive exams or just trying to get better at maths.

Table of Content

Definition
What are Factors of a Number?
Factors of Prime Numbers
Factors of Composite Numbers
Factors of Square Numbers
Formulas and Rules
Solved Examples
Practice Questions
Real Life Examples
Conclusion
FAQs on Factors of a Number

Definition

An integer that precisely divides a number without leaving a remainder is called a factor of the number. "B" is a factor of "A" if "A" is divisible by "B."

For example:

2 is a factor of 10 because 10 ÷ 2 = 5
3 is not a factor of 10 because 10 ÷ 3 = 3.33 (not an integer)

What are Factors of a Number?

All whole numbers that can be multiplied in pairs to return the original number are called factors of a number. There are at least two factors for every number (apart from 0): 1 and itself.

For example:
Factors of 12 are 1, 2, 3, 4, 6, and 12.
Explanation:
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12

So, all these numbers are factors of 12.

Factors of Prime Numbers

Prime numbers are special because they have only two factors:
1 and the number itself.

Examples:

Factors of 2: 1, 2
Factors of 3: 1, 3
Factors of 17: 1, 17

So, a prime number always has exactly two distinct factors.

Factors of Composite Numbers

Composite numbers are numbers that have more than two factors. These are numbers that can be divided by numbers other than 1 and itself.

Examples:

Factors of 8: 1, 2, 4, 8
Factors of 15: 1, 3, 5, 15

So, composite numbers always have more than two factors.

Factors of Square Numbers

A square number is a number that is the product of an integer multiplied by itself.

Examples:

36 = 6 × 6
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Observation:
Square numbers have an odd number of total factors because one of the factors repeats (e.g., 6 × 6).

Formulas and Rules

The following guidelines and formulas can be used to better understand factors:

Rule 1: One and itself are factors of every number.
The second rule states that a number can be divided by its factors.
Rule 3: Prime factorisation is used to determine the total number of factors.

Formula for total number of factors:
If N = a^p × b^q × c^r
Then, total number of factors = (p+1)(q+1)(r+1)

Example:
Find total factors of 36.
36 = 2² × 3²
Total factors = (2+1)(2+1) = 3 × 3 = 9 factors

Solved Examples

Example 1: Find factors of 18.
Factors: 1, 2, 3, 6, 9, 18

Example 2: Is 37 a prime number?
Yes, because it's only factors are 1 and 37.

Example 3: Find a total number of factors of 100.
100 = 2² × 5²
Total = (2+1)(2+1) = 3 × 3 = 9 factors

Example 4: List all factors of 48.
Factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Example 5: Find whether 121 is a perfect square and list its factors.
Yes, 121 = 11 × 11
Factors: 1, 11, 121

Example 6: Are 4 and 9 factors of 36?
Yes, because 36 ÷ 4 = 9 and 36 ÷ 9 = 4

Example 7: Find the sum of all factors of 20.
Factors: 1, 2, 4, 5, 10, 20
Sum = 42

Example 8: Write factors of 27.
Factors: 1, 3, 9, 27

Example 9: Find all prime factors of 60.
60 = 2² × 3 × 5
Prime factors: 2, 3, 5

Example 10: How many total factors does 72 have?
72 = 2³ × 3²
Total = (3+1)(2+1) = 4 × 3 = 12 factors

Practice Questions

Find all factors of 30
List the prime factors of 84
What are the factors of 16?
Determine whether 53 is prime or composite
Find the total number of factors of 225
Write down the factors of 100
Is 1 a factor of every number?
Find the common factors of 12 and 18
List all square numbers between 1 and 100 and find their factors
Write the factor pairs of 24

Real Life Examples

Tiling Floors: Only when the quantity of tiles used is proportional to the overall area can a floor be tiled with tiles of varying sizes.
Packaging: To ensure that nothing is left out, factors are taken into consideration when determining how many items fit in each box.
Time management is the process of allocating time in sessions, with minutes representing one hour.
Sports: Depending on the total number of players, players are divided into equal groups.
Music Beats: Recurring beats are frequently the basis of rhythmic patterns in music.

Conclusion

Numerous mathematical operations and practical applications are based on factors. We can better comprehend the structure and behaviour of numbers by dissecting them into their constituent factors. Perfect squares, prime numbers, and composite numbers all provide different perspectives on the realm of numbers.

Learning factors improves number sense and logic in addition to making equations simpler. The ability to work with factors opens the door to a deeper understanding of mathematics, whether one is solving puzzles, managing time, or even developing digital programs.

Related Links

Factor and Multiples - Master Factors and Multiples - Build a Strong Foundation in Number Theory with Easy Examples and Practice!

Highest Common Factor (HCF) - Crack HCF Problems Quickly - Learn Shortcuts, Tricks, and Real-Life Uses of Highest Common Factor!

Factors & Multiples - Types of Numbers - Explore Types of Numbers Through Factors & Multiples - Understand Primes, Composites, and More with Clear Explanations!

Factors, Common Factors, Multiples - Ace Factors, Common Factors, and Multiples - Grasp Core Concepts with Step-by-Step Solutions and Fun Problems!