CODEBHAVYA • MATHEMATICS

🔢 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?

Solution

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

Answer: Yes

Problem 2

Is 729 divisible by 9?

Solution

Sum of digits: 7 + 2 + 9 = 18.

18 is divisible by 9.

Answer: Yes

Problem 3

Find the HCF of 24 and 36.

Solution

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

Solve each problem first, then use Show Solution to check your method.

1. Is 246 divisible by 2?
2. Is 345 divisible by 5?
3. Find HCF of 18 and 24.
4. Find LCM of 8 and 12.
5. Is 729 divisible by 9?

📝 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.
INTERACTIVE PRACTICE

✍️ Number System Practice

You have learned, solved and revised the topic. Answer all 20 questions from easy to hard and submit once. Your score, correct answers and explanations appear after submission.

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