Number System Notes for Sub Engineer | Complete Guide

📘 Number System Notes (Sub Engineer)

🔢 Introduction

A number system is a method of representing numbers. It is the foundation of mathematics and plays a crucial role in competitive exams.

📊 Classification of Numbers

Natural Numbers: 1, 2, 3...
Whole Numbers: 0 included
Integers: Negative + Positive
Rational Numbers: p/q form
Irrational Numbers: √2, π
Real Numbers: All numbers combined

📌 Important Concepts

Prime Numbers: 2, 3, 5, 7
Composite Numbers: 4, 6, 8
Co-prime Numbers: HCF = 1
Even Numbers: Divisible by 2
Odd Numbers: Not divisible by 2

⚡ Divisibility Rules

2 → Last digit even
3 → Sum of digits divisible by 3
4 → Last 2 digits divisible by 4
5 → Ends in 0 or 5
9 → Sum divisible by 9
11 → Alternate sum difference

📘 HCF & LCM

LCM × HCF = Product of Numbers

HCF is the greatest divisor, and LCM is the smallest multiple.

🔵 Even and Odd Numbers

Even = 2n | Odd = 2n + 1

Addition

Operation	Result
Even + Even	Even
Odd + Odd	Even
Even + Odd	Odd

Subtraction

Operation	Result
Even − Even	Even
Odd − Odd	Even
Even − Odd	Odd
Odd − Even	Odd

Multiplication

Operation	Result
Even × Even	Even
Even × Odd	Even
Odd × Odd	Odd

📈 Important Sum Formulas

Sum of first n natural numbers = n(n+1)/2

Sum of first n even numbers = n(n+1)

Sum of first n odd numbers = n²

📝 Practice Questions

HCF of 36 and 48 = 12
LCM of 8 and 12 = 24
Sum of first 5 odd numbers = 25

LCM & HCF Complete Notes

LCM & HCF – Complete Notes for SSC, Railway, JE

📌 1. HCF (Highest Common Factor)

HCF is the largest number that divides two or more numbers exactly.

Example:

12 → 1,2,3,4,6,12
18 → 1,2,3,6,9,18

HCF = 6

📌 2. LCM (Least Common Multiple)

LCM is the smallest number which is a multiple of two or more numbers.

Example:

12 → 12, 24, 36...
18 → 18, 36...

LCM = 36

📌 3. Methods to Find HCF & LCM

✔ Prime Factorization Method

Example: 12, 18

12 = 2² × 3
18 = 2 × 3²

HCF = lowest powers → 2 × 3 = 6

LCM = highest powers → 2² × 3² = 36

✔ Division Method (HCF)

Example: 18, 12

18 = 12 × 1 + 6

12 = 6 × 2 + 0

HCF = 6

✔ Ladder Method (LCM)

Divide numbers by common primes until 1 remains. Multiply all divisors.

📌 4. Most Important Formula 🔥

HCF × LCM = Product of Numbers

Example: Numbers = 8, 12

HCF = 4

LCM = (8 × 12) / 4 = 24

📌 5. Special Cases

✔ Co-prime Numbers

HCF = 1
LCM = Product
Example: 8, 15 → LCM = 120

✔ One Number Divides Another

Example: 6, 24
HCF = 6
LCM = 24

📌 6. HCF of Fractions

HCF = (HCF of numerators) ÷ (LCM of denominators)

Example: 2/3, 4/9

HCF(2,4) = 2
LCM(3,9) = 9

Answer = 2/9

📌 7. LCM of Fractions

LCM = (LCM of numerators) ÷ (HCF of denominators)

Example: 2/3, 4/5

LCM(2,4) = 4
HCF(3,5) = 1

Answer = 4

📌 8. Important Points to Remember 🔥

HCF is always ≤ smallest number
LCM is always ≥ largest number
HCF divides all numbers
LCM is divisible by all numbers
Co-prime → HCF = 1
Simplify fractions before solving

📌 9. Exam Tricks 💯

Small numbers → use factors
Large numbers → use division method
Fractions → use direct formulas
Always check simplification

📌 10. Practice Questions

Find HCF of 24 and 36
Find LCM of 15 and 20
HCF = 5, Product = 150 → find LCM
Find HCF of 3/5 and 6/10
Find LCM of 2/3 and 4/5

📌 11. Quick Revision

HCF × LCM = Product of numbers
HCF (fractions) = HCF of numerators ÷ LCM of denominators
LCM (fractions) = LCM of numerators ÷ HCF of denominators

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