LCM - Least Common Multiple

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🔢 Great Common Multiple - Basics

The smallest number that is exactly divisible by two or more!

📚 What is the least common multiple?

Least Common Multiple (LCM)is exactly divisible by two or more numbersThe smallest numberis

🎯 Simple explanation:
• LCM is the least common multiple of the given numbers
• LCM is always greater than or equal to the given numbers
• Each number will exactly divide the LCM

⭐ Basic concepts

Comment	Explanation	example
Multiple	A number obtained by multiplying a number by a whole number	Multiples of 4: 4, 8, 12, 16, 20...
Common multiple	A number that is a multiple of two numbers	Common multiples of 4, 6: 12, 24, 36...
The remainder is a common multiple	is the smallest common multiple	LCM(4, 6) = 12

📐 LCM diagnostic methods

1️⃣ Factorization method

Prime Factorization Method

For simple numbers

2️⃣ Division method

Division Method

For multiple numbers

3️⃣ Using HCF

Using HCF Formula

For two numbers

🔢 Method 1: Prime Factorization

Steps:

Write each number as a product of prime numbers
Take all the weather factors
Take maximum layers
Multiply them

Example: LCM(12, 18)
12 = 2² × 3¹
18 = 2¹ × 3²
All factors: 2 and 3
Maximum layer: 2² × 3² = 4 × 9 =36

🔢 Method 2: Division Method

Steps:

Write all the numbers in a row
Divide any number by the divisor prime number
Write the non-divisible numbers as they are
Continue until all are 1
Multiply the divided numbers

Example: LCM(12, 15, 20)
2 | 12, 15, 20
2 | 6, 15, 10
3 | 3, 15, 5
5 | 1, 5, 5
| 1, 1, 1
LCM = 2 × 2 × 3 × 5 =60

📊 Key features

trait	Explanation
LCM(a, b) ≥ max(a, b)	LCM is always greater than or equal to the larger number
LCM(a, a) = a	LCM of a number is that number
LCM(a, 1) = a	LCM with any number 1 = that number
LCM(a, b) = a × b / HCF(a,b)	LCM can be detected if HCF is known
LCM = a×b if a and b are independent	LCM = product of co-prime numbers

🎯 LCM applications

Find out the time of the bell
orbital rendezvous time
The time when the lights are on simultaneously
Common Denominator of Fractions
Find equal measurements

📐 Great Common Multiplier - Key Formulas

Most Important LCM Formulas for TNPSC Exam!

🔢 Basic formulas

formula	Explanation
LCM × HCF = a × b	LCM × HCF of two numbers = Product of those two numbers
LCM = (a × b) / HCF	LCM can be detected if HCF is known
HCF = (a × b) / LCM	HCF can be found if LCM is known
LCM(a, b, c) = LCM(LCM(a,b), c)	Calculate LCM of three numbers step by step

📊 LCM of fractions

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

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

🎯 Divisional Formulas

Question type	formula
The smallest number that is exactly divisible by a, b, c	LCM(a, b, c)
The smallest number that gives a remainder when divided by a, b, c and x, y, z respectively	LCM(a, b, c) − k(k = a-x = b-y = c-z)
The largest number that gives the remainder when divided by a, b, c respectively x, y, z	LCM(a, b, c) + k(k general remainder)

📐 Special formulas

Type	formula	example
Co-prime numbers	LCM = a × b	LCM(7, 11) = 7 × 11 = 77
One is a multiple of the other	LCM = Large Number	LCM(5, 15) = 15
Sequence numbers (n, n+1)	LCM = n × (n+1)	LCM(8, 9) = 72

⏰ Time & Cycle formulas

Question type	formula
a, b, c are the times in seconds for the bell to strike simultaneously	LCM(a, b, c) sec
A, b, c is the point of intersection of the paths of radius m	LCM(a, b, c) after meter travel
Lamps a, b, c flash in seconds, time of simultaneous flashing	LCM(a, b, c) sec

📊 HCF and LCM relationship

trait	Explanation
HCF ≤ LCM	HCF is always less than or equal to LCM
HCF is a factor of 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)
HCF × LCM = a × b	Applies to two numbers only

📐 Key formula set

For two numbers:
• LCM × HCF = a × b
• LCM = (a × b) / HCF
• a = LCM × HCF / b

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

📝 Great Common Multiplier - 10 Key Examples

TNPSC Model Questions with Step by Step Solutions!

📌 Example 1: Factorization method

Question:LCM(16, 24, 36) = ?

Solution (Prime Factorization):
16 = 2⁴
24 = 2³ × 3¹
36 = 2² × 3²
All factors: 2 and 3
Maximum layer: 2⁴ × 3² = 16 × 9 =144

Answer: 144

📌 Example 2: Division method

Question:LCM(12, 18, 20) = ?

Solution (Division Method):
2 | 12, 18, 20
2 | 6, 9, 10
3 | 3, 9, 5
3 | 1, 3, 5
5 | 1, 1, 5
| 1, 1, 1
LCM = 2 × 2 × 3 × 3 × 5 =180

Answer: 180

📌 Example 3: LCM of fractions

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

Solution:
Formula: LCM = LCM(Volumes) / HCF(Areas)
LCM(2, 4, 6) = 12
HCF(3, 5, 7) = 1
LCM = 12/1 =12

Answer: 12

📌 Example 4: LCM using HCF

Question:If the product of two numbers is 1800, HCF = 15, then LCM = ?

Solution:
Formula: HCF × LCM = a × b
15 × LCM = 1800
LCM = 1800 / 15 =120

Answer: 120

📌 Example 5: The bell ringing question

Question:Three bells ring at 4, 6, and 8 seconds respectively. If they hit simultaneously, how many seconds will it take for them to hit simultaneously again?

Solution:
Hit Simultaneously = LCM(4, 6, 8)
4 = 2²
6 = 2 × 3
8 = 2³
LCM = 2³ × 3 = 8 × 3 =24 seconds

Answer: 24 seconds

📌 Example 6: Remaining question

Question:What is the smallest number which when divided by 4, 5, 6 gives a remainder of 2, 3, 4 respectively?

Solution:
4 - 2 = 2
5 - 3 = 2
6 - 4 = 2
Common difference = 2 (k)
Number = LCM(4, 5, 6) - k
LCM(4, 5, 6) = 60
Number = 60 - 2 =58

Answer: 58

📌 Example 7: Orbit Question

Question:A, B, C complete one round of the runway in 3, 4, 6 minutes respectively. If we start from the same point, how many minutes will it take to meet again at that point?

Solution:
Meeting Time = LCM(3, 4, 6)
3 = 3
4 = 2²
6 = 2 × 3
LCM = 2² × 3 = 4 × 3 =12 minutes

Answer: 12 minutes

📌 Example 8: Finding Numbers

Question:HCF of two numbers = 9, LCM = 270, if one number is 27 then other number?

Solution:
Formula: HCF × LCM = a × b
9 × 270 = 27 × b
2430 = 27 × b
b = 2430 / 27 =90

Answer: 90

📌 Example 9: Lamp question

Question:The three lights flash once every 24, 36, 48 seconds respectively. If it flashes simultaneously at 8 AM, when will it next flash simultaneously?

Solution:
LCM(24, 36, 48)
24 = 2³ × 3
36 = 2² × 3²
48 = 2⁴ × 3
LCM = 2⁴ × 3² = 16 × 9 = 144 seconds
144 seconds = 2 minutes 24 seconds
Next time =8:02:24 AM

Answer: 8:02:24 AM

📌 Example 10: TNPSC Model

Question:If sum of two numbers HCF = 8, LCM = 96 is 40, what are they?

Solution:
Numbers = 8m, 8n (m, n are independent)
LCM = 8mn = 96 → mn = 12
Sum: 8m + 8n = 40 → m + n = 5
m + n = 5, mn = 12
(m, n) = (3, 4) or (4, 3)
Numbers = 8×3, 8×4 =24, 32

Answer: 24 and 32

💡 Important Notes:

Take maximum plots to find LCM
Take minimum plots to find HCF
LCM is used for bell, lamp, circuit questions
The formula HCF × LCM = a × b is very important

⚡ Reciprocal General Multiplication - Shortcuts & Tricks

Use These Shortcuts to Save Time in TNPSC Exam!

🚀 Shortcut 1: LCM of special numbers

Number type	LCM	example
Sequence numbers (n, n+1)	n × (n+1)	LCM(9, 10) = 90
Co-prime numbers	a × b	LCM(7, 11) = 77
One is a multiple of the other	Big number	LCM(6, 18) = 18
Same number, same number	That number	LCM(15, 15) = 15

🚀 Shortcut 2: LCM = (a × b) / HCF

Rule:	If HCF is known, LCM = product / HCF
Example:	LCM(24, 36) = ? HCF(24, 36) = 12 LCM = (24 × 36) / 12 = 864/12 =72

🚀 Shortcut 3: Quick Factorization

No	Factors
12	2² × 3
18	2 × 3²
24	2³ × 3
36	2² × 3²
48	2⁴ × 3
60	2² × 3 × 5
72	2³ × 3²
96	2⁵ × 3
100	2² × 5²
120	2³ × 3 × 5

🚀 Shortcut 4: Remaining questions

Question Type 1:	When divided by a, b, c respectively (a-k), (b-k), (c-k) remainder (k common)
Shortcut:	LCM(a, b, c) − k
Example:	When divided by 5, 6, 7 the remainder is 3, 4, 5 k = 5-3 = 6-4 = 7-5 = 2 Number = LCM(5,6,7) - 2 = 210 - 2 =208

🚀 Shortcut 5: Fractional LCM shortcut method

Formula:	LCM(a/b,c/d) = LCM(a,c) / HCF(b,d)
Example:	LCM(3/4, 5/6) = LCM(3,5) / HCF(4,6) = 15 / 2 =15/2

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

🚀 Shortcut 6: LCM from 1 to n

n	LCM(1 to n)
1 to 5	60
1 to 6	60
1 to 7	420
1 to 8	840
1 to 9	2520
1 to 10	2520

🚀 Shortcut 7: Comparison of HCF and LCM

HCF	LCM
Minimum layer	Maximum layer
General factors only	All factors
A smaller or equal number	A greater or equal number
Division questions	Multiple questions

🚀 Shortcut 8: For three numbers

Rule:	LCM(a, b, c) = LCM(LCM(a,b), c)
Example:	LCM(4, 6, 8) Step 1: LCM(4, 6) = 12 Step 2: LCM(12, 8) =24

🚀 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 LCM = hmn → mn = l/h
Example:	HCF=6, LCM=72, Sum=42 m + n = 42/6 = 7 mn = 72/6 = 12 m=3, n=4 → Numbers =18, 24

🚀 Shortcut 10: Time & Round questions

Question type	Answer
bell/lamp at the same time	LCM(Intervals)
Meeting on the runway	LCM(Cycle Times)
How many times meet (in time T)	T / LCM

Example:A bell rings at intervals of 3, 4, 5 minutes, how many times it rings together in 1 hour?
LCM(3,4,5) = 60 min
1 hour = 60 minutes
Junction = 60/60 =1 time(+ start = 2 times)

🎯 TNPSC Exam Tips:

Memorize the factorization table
Write the LCM of consecutive numbers, independent numbers directly
Use the formula HCF × LCM = a × b frequently
Find the value of k in the remaining questions
Do back calculation from Options

⏱️ Time Management:

For small numbers multiply directly and divide by HCF
Division method is fast for many numbers
Do not spend more than 45 seconds on a question

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