HCF - Highest Common Factor

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🔢 Meperu Common Factor - Basics

The largest number that can divide two or more numbers!

📚 What is Meperu Common Factor?

Highest Common Factor (HCF)is exactly divisible by two or more numbersThe largest numberis

🎯 Other Names:
• GCD - Greatest Common Divisor
• GCF - Greatest Common Factor
• HCF - Highest Common Factor

⭐ Basic concepts

Comment	Explanation	example
Factor	A number that exactly divides a number	Factors of 12: 1, 2, 3, 4, 6, 12
General factor	A number that divides two numbers	Common factors of 12, 18 are: 1, 2, 3, 6
More common factor	The largest of the common factors	HCF(12, 18) = 6

📐 HCF detection methods

1️⃣ Factorization method

Prime Factorization Method

For simple numbers

2️⃣ Division method

Division Method

For larger numbers

3️⃣ Euclid's method

Euclidean Algorithm

For larger numbers

🔢 Method 1: Prime Factorization

Steps:

Write each number as a product of prime numbers
Identify common predisposing factors
Take minimum layers
Multiply them

Example: HCF(24, 36)
24 = 2³ × 3¹
36 = 2² × 3²
General Factors: 2 and 3
Lower layer: 2² × 3¹ = 4 × 3 =12

🔢 Method 2: Division Method

Steps:

Divide the larger number by the smaller number
Divide the remainder by division
Continue until the remainder is 0
Last denominator = HCF

Example: HCF(48, 18)
48 ÷ 18 = 2, remainder = 12
18 ÷ 12 = 1, remainder = 6
12 ÷ 6 = 2, remainder = 0
HCF =6

📊 Key features

trait	Explanation
HCF(a, b) ≤ min(a, b)	HCF is always less than or equal to the smallest number
HCF(a, a) = a	The HCF of a number is that number
HCF(a, 1) = 1	HCF = 1 with any number 1
HCF(a, 0) = a	HCF with zero = that number
HCF = 1 if a and b are independent	HCF of co-prime numbers = 1

🎯 HCF applications

To simplify fractions
Cut into equal pieces
Divide into equal groups
Find the largest equal quantity
To simplify math problems

📐 Multiplicative Common Factor - Key Formulas

Most Important HCF Formulas for TNPSC Exam!

🔢 Basic formulas

formula	Explanation
HCF × LCM = a × b	HCF × LCM of two numbers = Product of those two numbers
LCM = (a × b) / HCF	If HCF is known to detect LCM
HCF = (a × b) / LCM	If LCM is known to detect HCF
HCF(a, b, c) = HCF(HCF(a,b), c)	Calculate HCF of three numbers step by step

📊 HCF of fractions

Type	formula	example
HCF of fractions	HCF = HCF(Volumes) / LCM(Parts)	HCF(2/3, 4/5) = HCF(2,4)/LCM(3,5) = 2/15
LCM of fractions	LCM = LCM(Volumes) / HCF(Parts)	LCM(2/3, 4/5) = LCM(2,4)/HCF(3,5) = 4/1 = 4

💡 Remember:
HCF - HCF of blocks, LCM of parts
LCM - LCM of blocks, HCF of parts

🎯 Divisional Formulas

Question type	formula
The largest number that divides a, b, c and gives an equal remainder	HCF(a-b, b-c, c-a)
When dividing a, b, c respectively x, y, z is the largest number that gives remainder	HCF(a-x, b-y, c-z)
The largest number that exactly divides a, b, c	HCF(a, b, c)

📐 Euclidean Algorithm

Formula:	HCF(a, b) = HCF(b, a mod b) when a > b
Stop condition:	HCF(a, 0) = a

Example: HCF(252, 105)
HCF(252, 105) = HCF(105, 252 mod 105) = HCF(105, 42)
HCF(105, 42) = HCF(42, 105 mod 42) = HCF(42, 21)
HCF(42, 21) = HCF(21, 42 mod 21) = HCF(21, 0)
HCF =21

🔢 HCF of serial numbers

Number type	HCF	example
Sequence numbers (n, n+1)	1	HCF(7, 8) = 1
A series of odd numbers	1	HCF(15, 17) = 1
Consecutive even numbers	2	HCF(14, 16) = 2
Co-prime numbers	1	HCF(8, 15) = 1

📊 HCF and LCM relationship

trait	Explanation
HCF ≤ LCM	HCF is always less than or equal to LCM
HCF is a factor of both LCM	HCF always divides LCM
LCM is a multiple of HCF	LCM is always a multiple of HCF
If HCF = LCM	Both numbers are equal (a = b)

📐 Key formula set

For two numbers:
• HCF × LCM = a × b
• a = HCF × m
• b = HCF × n
• (m, n are independent)

For fractions:
• HCF = HCF(volume) / LCM(area)
• LCM = LCM(volume) / HCF(area)

📝 Meperu Common Factor - 10 Key Examples

TNPSC Model Questions with Step by Step Solutions!

📌 Example 1: Factorization method

Question:HCF(24, 36, 48) = ?

Solution (Prime Factorization):
24 = 2³ × 3¹
36 = 2² × 3²
48 = 2⁴ × 3¹
General Factors: 2 and 3
Lower layer: 2² × 3¹ = 4 × 3 =12

Answer: 12

📌 Example 2: Division method

Question:HCF(56, 98) = ?

Solution (Division Method):
98 ÷ 56 = 1, remainder = 42
56 ÷ 42 = 1, remainder = 14
42 ÷ 14 = 3, remainder = 0
HCF =14

Answer: 14

📌 Example 3: HCF of fractions

Question:HCF(2/3, 4/5, 6/7) = ?

Solution:
Formula: HCF = HCF(Volumes) / LCM(Parts)
HCF(2, 4, 6) = 2
LCM(3, 5, 7) = 105
HCF = 2/105 =2/105

Answer: 2/105

📌 Example 4: HCF × LCM formula

Question:If the product of two numbers is 2160, HCF = 12, then LCM = ?

Solution:
Formula: HCF × LCM = a × b
12 × LCM = 2160
LCM = 2160 / 12 =180

Answer: 180

📌 Example 5: Equal remainder question

Question:54, 87, 120 which is the largest number which gives the same remainder when divided?

Solution:
Formula: HCF(a-b, b-c, c-a)
87 - 54 = 33
120 - 87 = 33
120 - 54 = 66
HCF(33, 33, 66) =33

Answer: 33

📌 Example 6: Specified remainder

Question:65, 117, 181 when divided by 5, 9, 13 is the largest number which gives remainder respectively?

Solution:
Formula: HCF(a-x, b-y, c-z)
65 - 5 = 60
117 - 9 = 108
181 - 13 = 168
HCF(60, 108, 168):
60 = 2² × 3 × 5
108 = 2² × 3³
168 = 2³ × 3 × 7
HCF = 2² × 3 =12

Answer: 12

📌 Example 7: Even Pieces

Question:Cut 3 wires of length 84 cm, 126 cm, 168 cm into equal pieces. What is the maximum length of a piece?

Solution:
maxLength = HCF(84, 126, 168)
84 = 2² × 3 × 7
126 = 2 × 3² × 7
168 = 2³ × 3 × 7
HCF = 2 × 3 × 7 =42 cm

Answer: 42 cm

📌 Example 8: Finding two numbers

Question:HCF = 12, LCM = 180 of two numbers, if one number is 36 then other number?

Solution:
Formula: HCF × LCM = a × b
12 × 180 = 36 × b
2160 = 36 × b
b = 2160 / 36 =60

Answer: 60

📌 Example 9: Euclid's Method

Question:HCF(1071, 462) = ? (in Euclid's method)

Solution:
1071 = 462 × 2 + 147
462 = 147 × 3 + 21
147 = 21 × 7 + 0
HCF =21

Answer: 21

📌 Example 10: TNPSC Model

Question:If the sum of two numbers HCF = 6, LCM = 180 is 66, what are they?

Solution:
Numbers = 6m, 6n (m, n are independent)
LCM = 6mn = 180 → mn = 30
Sum: 6m + 6n = 66 → m + n = 11
m + n = 11, mn = 30 → m = 5, n = 6
Numbers = 6×5, 6×6 =30, 36

Answer: 30 and 36

💡 Important Notes:

The factorization method is simple for small numbers
The division method is better for larger numbers
The formula HCF × LCM = a × b is very important
Find the HCF of the differences in equal remainder questions

⚡ Meeperu General Factor - Shortcuts & Tricks

Use These Shortcuts to Save Time in TNPSC Exam!

🚀 Shortcut 1: HCF of serial numbers

Number type	HCF	example
Sequence numbers (n, n+1)	Always 1	HCF(99, 100) = 1
n, n+2 (double space)	1 or 2	HCF(10, 12) = 2
A series of odd numbers	Always 1	HCF(33, 35) = 1
Consecutive even numbers	Always 2	HCF(24, 26) = 2

🚀 Shortcut 2: Division shortcut

Rule:	Subtract the smaller number from the larger number and repeat with the remainder
Example:	HCF(48, 18) 48 - 18 = 30 30 - 18 = 12 18 - 12 = 6 12 - 6 = 6 6 - 6 = 0 → HCF =6

🚀 Shortcut 3: Numbers divisible by 2, 3, 5

No	Divisibility rule
2	If the last digit is 0, 2, 4, 6, 8
3	If the sum of the digits is divisible by 3
4	If the last two digits are divisible by 4
5	If the last digit is 0 or 5
6	If divisible by both 2 and 3
8	If the last three digits are divisible by 8
9	If the sum of digits is divisible by 9
10	If the last digit is 0
11	Difference of single and double places is 0 or 11 times

🚀 Shortcut 4: Application of HCF × LCM

Formula:	HCF × LCM = a × b
Question:	HCF = 8, one number = 24, LCM = 120, other number = ?
Shortcut:	Other number = (HCF × LCM) / first number = (8 × 120) / 24 = 960/24 =40

🚀 Shortcut 5: Equal remainder question

Question Type:	The largest number that gives an equal remainder when dividing a, b, c
Shortcut:	HCF(b-a, c-b, c-a)
Example:	a=35, b=56, c=91 56-35=21, 91-56=35, 91-35=56 HCF(21, 35, 56) =7

🚀 Shortcut 6: Fractional HCF shortcut method

Formula:	HCF(a/b,c/d) = HCF(a,c) / LCM(b,d)
Example:	HCF(3/4, 5/6) = HCF(3,5) / LCM(4,6) = 1 / 12 =1/12

💡 Remember:For HCF – Volumes are HCF, Areas are LCM

🚀 Shortcut 7: Base number HCF

numbers	HCF	Explanation
Any number, 1	1	1 divides everything
independent numbers	1	There is no common factor
Same number, same number	That number	HCF(15,15) = 15
Number is its multiple	Small number	HCF(5,15) = 5

🚀 Shortcut 8: For three numbers

Rule:	HCF(a, b, c) = HCF(HCF(a,b), c)
Example:	HCF(24, 36, 48) Step 1: HCF(24, 36) = 12 Step 2: HCF(12, 48) =12

🚀 Shortcut 9: Find numbers

Information:	HCF = h, LCM = l, Sum = s
Method:	Numbers = hm, hn (m, n are independent) hm + hn = s → m + n = s/h hmn × h = l → mn = l/h²
Example:	HCF=4, LCM=48, Sum=28 m + n = 28/4 = 7 mn = 48/4 = 12 m=3, n=4 → Numbers =12, 16

🚀 Shortcut 10: Quick test

Test	Explanation
HCF verification	HCF always divides two numbers exactly
LCM validation	LCM should always be multiple of HCF
Multiplicity check	Check that HCF × LCM = a × b is correct

🎯 TNPSC Exam Tips:

Find factors quickly by remembering the division rules
Write the HCF of consecutive numbers directly as 1 or 2
The formula HCF × LCM = a × b is useful for many questions
Find the HCF of the differences in equal remainder questions
Do back calculation from Options

⏱️ Time Management:

Find common factors directly for small numbers
The division method is faster for larger numbers
Do not spend more than 45 seconds on a question

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