LCM (Least Common Multiple)

Definition →The smallest number that is divisible by all given numbers.

Steps to Find LCM
Method 1 – Prime Factorization

• Factorize each number into prime numbers.

• Take each prime factor only once with its highest power.

• Multiply them together.

Example:
Find LCM of 12 and 18.

12 = 2² × 3¹
18 = 2¹ × 3²

LCM = 2² × 3² = 36

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Method 2 – Division Method (Shortcut)

• Divide the numbers by prime numbers till all become 1.

• Multiply all divisors together.

Example:
Find LCM of 15, 20, 30.

Divisor: 15 , 20, 30

2- 15 10 15
2- 15 5 15
3- 5 5 5
5- 1 1 1

LCM = 2 × 2 × 3 × 5 = 60

HCF (Highest Common Factor)
It is also called GCD (Greatest Common Divisor).

#Definition
The HCF of two or more numbers is the largest positive integer that divides each of the numbers completely (without leaving any remainder).

Steps to Find HCF:

Method 1: Prime Factorization

1. Find the prime factors of each number.

2. Identify the common prime factors.

3. Multiply them to get the HCF.

Example:
Find HCF of 18 and 24.

Prime factors of 18 → 2×3×3
Prime factors of 24 → 2×2×2×3

Common factors → 2×3-> HCF = 6

Method 2: Division Method (Euclidean Algorithm)

1. Divide the larger number by the smaller number.

2. Take the remainder.

3. Divide the smaller number by the remainder.

4. Repeat until the remainder becomes 0.
The last divisor is the HCF.

Example:
Find HCF of 36 and 60:

60/36=1 remainder 24
36/24=1 remainder 12
24/12=2 remainder 0

HCF = 12

Another way to find HCF when Two numbers are there and LCM is Given :

Formula: HCF × LCM = N1 × N2

e.g: N1 = 15, N2 = 20, LCM = 60, HCF= ?

HCF = 15×20/60
= 300/60
HCF = 5


Note: This formula is valid for only two numbers.

Face value And Place value
1. Face Value
The face value of a digit is the actual value of the digit itself, irrespective of its position in the number.

#Example
For the number 4725:
Face value of 4 = 4
Face value of 7 = 7
Face value of 2 = 2
Face value of 5 = 5
✅ Face value = digit itself

2. Place Value
It is the place at which Digit located like 10th position, 100th position etc.

#Example:
For the number 4725:
Place value of 4 = 4000
Place value of 7 = 700
Place value of 2 = 20
Place value of 5 = 5

Quick Trick to Remember
Face value → What you see 👀
Place value → Where it is 📍

Types of Fractions

1. Proper Fraction:
• A Fraction where the Numerator < denominator.
• Value of fraction is less than 1.
• Example: 3/5, 6/7, 9/11.

2. Improper Fraction:
• A Fraction where the Numerator > denominator.
• Value of fraction is greater than 1.
•Example: 8/5, 7/3, 11/7.

3. Mixed Fraction:
    • A fraction that is combination of a whole number and a proper fraction.
    • Every mixed fraction can be converted into an improper fraction, and vice versa
•Example: 2(1/3) = 7/3,
5(2/4) = 22/4.

Fractions Decimals Percentage
1/2 0.5 50%
1/4 0.25 25%
1/5 0.20 20%
1/10 0.10 10%
1/20 0.05 5%
1/8 0.125 12.5%
3/4 0.75 75%
4/5 0.80 80%
1/3 0.3333 33.33%
2/3 0.6666 66.66%
1/6 0.1666 16.66%
5/6 0.8333 83.33%
6/5 1.2 120%
5/4 1.25 125%
3/2 1.5 150%
1/7 0.1428 14.28%
1/9 0.1111 11.11%
1/11 0.0909 9.09%
1/12 0.0833 8.33%
1/15 0.0666 6.66%
1/16 0.0625 6.25%

Comparing Fractions

1. Same Denominator
If fractions have the Same Denominator, Just compare numerators.
Example: 5/9 < 7/9

2. Same Numerator
If fractions have the same numerator, the one with the smaller denominator is larger.
Example: 3/5 > 3/7

3. Different Numerator and Denominator
There are two methods:

i. Cross Multiplication
a/b and c/d is there,
check a×d and c×b,
If a×d > c×b then a/b > c/d,
else c/d > a/b
Example:
3/5 and 4/8, check 3×8 and 4×5, i.e. 24 > 20, 3/5 > 4/8.

ii. Convert to same Denominator
Example:
5/7 and 3/4, 5/7 × 4/4 and 3/4 × 7/7, i.e 20/28 and 21/28
Therefore, we can say that,
5/7 < 3/5.

Percentage Increase & Decrease:

1. Percentage increase:
When the value goes up, use,
Percentage increase= ((New value - Old value) ÷ Old value) × 100

Example:
Old price = 100
New price = 120
Increase = 120 − 100 = 20
Percentage increase = (20 / 100) × 100 = 20% Increase

2. Percentage decrease:
          When the value goes down, use,
Percentage decrease (Old value - New value) ÷ Old value) × 100

Example:
Old price = 100
New price = 70
Decrease = 100 − 70 = 20
Percentage decrease = (30 / 100) × 100 = 30% Decrease

Shortcut Trick:
-Increase → New − Old
-Decrease → Old − New
-Always divide by Old value & multiply by 100.

1. 20 is 25% of 80. So, 80 getting 100 means 25% increase in 80.
Don't be in a hurry, Solve with formulas.
2. 100-80=20 increase
(20÷(80 old value))×100 = 25%