Factors, Multiples & Prime Numbers: A Complete Guide

Introduction

Have you ever wondered why some numbers are called prime, while others have multiple factors? Understanding factors, multiples, and prime numbers is essential in mathematics and is widely used in cryptography, coding, and problem-solving.

In this guide, you’ll learn:

What factors and multiples are
The difference between prime and composite numbers
How to find factors, multiples, and prime numbers
Real-life applications of these concepts

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Understanding Factors

A factor of a number is a number that divides it exactly (without leaving a remainder).

Example: Factors of 12 → 1, 2, 3, 4, 6, 12
(Each of these numbers divides 12 without a remainder.

How to Find Factors of a Number?

To find factors of a number:

Start with 1 and the number itself.
Check for numbers that divide exactly.
List all such numbers in pairs.

Example:
Find the factors of 18.
Step 1: Start with 1 and 18.
Step 2: Check for other divisors: 2, 3, 6, and 9.
Step 3: Write them in pairs → 1, 2, 3, 6, 9, 18.

Understanding Multiples

A multiple of a number is the result of multiplying it by whole numbers.

Example:
Multiples of 5 → 5, 10, 15, 20, 25, 30, …
(These numbers are all obtained by multiplying 5 by 1, 2, 3, etc.)

How to Find Multiples of a Number?

To find multiples of a number:
Multiply the number by 1, 2, 3, 4…
List the first few results.

Example:
Find the first 5 multiples of 7.
Step 1: Multiply 7 × 1 = 7
Step 2: Multiply 7 × 2 = 14
Step 3: Multiply 7 × 3 = 21
Step 4: Multiply 7 × 4 = 28
Step 5: Multiply 7 × 5 = 35
Multiples of 7 → 7, 14, 21, 28, 35

Prime and Composite Numbers

What Are Prime Numbers?

A prime number is a number that has exactly two factors: 1 and itself.

Examples:
Prime numbers → 2, 3, 5, 7, 11, 13, 17, …
(They have only two factors: 1 and the number itself.)

What Are Composite Numbers?

A composite number is a number that has more than two factors.

Examples:
Composite numbers → 4, 6, 8, 9, 10, 12, 14, …
(Factors of 6 → 1, 2, 3, and 6.)

Finding Prime Numbers Using the Sieve of Eratosthenes

The Sieve of Eratosthenes is a simple way to find prime numbers up to a certain limit.

Steps to Use the Sieve:

Write numbers from 1 to N.
Cross out 1 (not prime).
Start with 2 and cross out all its multiples.
Move to the next uncrossed number and repeat.
The remaining numbers are prime.

Example:
To find prime numbers up to 30, follow the steps above and you get:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Prime Factorization: The Building Blocks of Numbers

Prime factorization is breaking a number down into prime numbers that multiply to give the original number.

Example:
Find the prime factorization of 36.
Step 1: Start dividing by the smallest prime (2):
36 ÷ 2 = 18
18 ÷ 2 = 9
Step 2: Now divide by 3 (next prime number):
9 ÷ 3 = 3
3 ÷ 3 = 1

Prime factorization of 36 = 2 × 2 × 3 × 3 or 2² × 3².

Lowest Common Multiple (LCM) & Highest Common Factor (HCF)

Finding the LCM

The LCM of two numbers is the smallest multiple they both share.

Example:
Find the LCM of 4 and 6.
Multiples of 4 → 4, 8, 12, 16, 20
Multiples of 6 → 6, 12, 18, 24
LCM = 12 (smallest number common in both lists).

Finding the HCF

The HCF of two numbers is the largest factor they both share.

Example:
Find the HCF of 18 and 24.
Factors of 18 → 1, 2, 3, 6, 9, 18
Factors of 24 → 1, 2, 3, 4, 6, 8, 12, 24
HCF = 6 (largest common factor).