🔢 Number System

Learn the different types of numbers, their properties, divisibility rules, factors, multiples, HCF and LCM.

🎯 Learning Objectives

After completing this topic, you should be able to:

  • Identify different types of numbers.
  • Understand number properties.
  • Apply divisibility rules.
  • Find factors and multiples.
  • Calculate HCF and LCM.
  • Solve number-system problems quickly.

📖 1. What is a Number System?

A number system is a method of representing and classifying numbers.

In mathematics, numbers are classified into different categories based on their properties.

🔢 2. Types of Numbers

Natural Numbers

Natural numbers are positive counting numbers.

N = {1, 2, 3, 4, 5, ...}

Whole Numbers

Whole numbers include zero and all natural numbers.

W = {0, 1, 2, 3, 4, ...}

Integers

Integers include positive numbers, negative numbers and zero.

Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Rational Numbers

A number that can be expressed in the form p/q, where q ≠ 0.

Rational Number = p / q

Irrational Numbers

Numbers that cannot be expressed as p/q are called irrational numbers.

Examples: √2, √3, π

Real Numbers

Real numbers include both rational and irrational numbers.

🔐 3. Prime and Composite Numbers

Prime Number

A number having exactly two positive factors: 1 and itself.

Example:

7 has factors: 1 and 7

Therefore, 7 is prime.

Composite Number

A number having more than two positive factors is called composite.

Example:

12 has factors: 1, 2, 3, 4, 6, 12

Therefore, 12 is composite.

Important: 1 is neither prime nor composite.

📌 4. Divisibility Rules

Divisibility by 2
Last digit must be even: 0, 2, 4, 6 or 8.
Divisibility by 3
Sum of digits must be divisible by 3.
Divisibility by 5
Last digit must be 0 or 5.
Divisibility by 9
Sum of digits must be divisible by 9.
Divisibility by 10
Last digit must be 0.

🔗 5. Factors and Multiples

Factors

Factors are numbers that divide another number exactly without leaving a remainder.

Factors of 12:

1, 2, 3, 4, 6, 12

Multiples

Multiples are obtained by multiplying a number by 1, 2, 3, ...

Multiples of 5:

5, 10, 15, 20, 25, ...

🔵 6. HCF

HCF means Highest Common Factor.

HCF = Greatest number that divides all given numbers.

Example

Find HCF of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

HCF = 6

🟢 7. LCM

LCM means Least Common Multiple.

LCM = Smallest positive number divisible by all given numbers.

Example

Find LCM of 4 and 6.

Multiples of 4: 4, 8, 12, 16, ...

Multiples of 6: 6, 12, 18, ...

LCM = 12

📐 8. Important Relationship

HCF × LCM = Product of the two numbers
For 12 and 18:

HCF = 6
LCM = 36

6 × 36 = 216
12 × 18 = 216

Therefore, the relationship is verified.

🧮 9. Solved Problems

Problem 1

Is 135 divisible by 5?

The last digit is 5. Therefore, it is divisible by 5.

Answer: Yes

Problem 2

Is 729 divisible by 9?

Sum of digits: 7 + 2 + 9 = 18

18 is divisible by 9.

Answer: Yes

Problem 3

Find the HCF of 24 and 36.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

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

HCF = 12

⚡ 10. Quick Tricks

For divisibility by 2, check only the last digit.
For divisibility by 5, last digit must be 0 or 5.
For divisibility by 9, add all digits.
If a number is divisible by both 2 and 3, it is divisible by 6.

✍️ 11. Practice Problems

Ready to test yourself? Try all 20 questions, from basic to advanced.

🚀 Start 20-Question Practice
1. Is 246 divisible by 2?
Last digit = 6.
6 is even.
Answer: Yes
2. Is 345 divisible by 5?
Last digit = 5.
Therefore it is divisible by 5.
Answer: Yes
3. Find 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
4. Find LCM of 8 and 12.
Multiples of 8: 8, 16, 24, 32, ...
Multiples of 12: 12, 24, 36, ...
LCM = 24
5. Is 729 divisible by 9?
Sum of digits: 7 + 2 + 9 = 18
18 is divisible by 9.
Answer: Yes

📝 Quick Revision

  • Natural numbers start from 1.
  • Whole numbers include 0.
  • Integers include positive, negative and zero.
  • Prime numbers have exactly two factors.
  • 1 is neither prime nor composite.
  • HCF means Highest Common Factor.
  • LCM means Least Common Multiple.
  • HCF × LCM = Product of two numbers.