SSC GD LCM and HCF: Concepts, Formulas, Tricks and Examples

LCM (Least Common Multiple) and HCF (Highest Common Factor) are among the most important topics in SSC GD Mathematics. Every year, multiple questions are asked directly or indirectly from these concepts. Many students lose easy marks because they do not understand when to use LCM and when to use HCF.

What is HCF (Highest Common Factor)?

The HCF of two or more numbers is the greatest number that divides all the given numbers completely.

Example

HCF of 12 and 18
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18

Common factors: 1, 2, 3, 6
Greatest factor = 6 (HCF)

What is LCM (Least Common Multiple)?

The LCM of two or more numbers is the smallest number that is completely divisible by all given numbers.

Example

LCM of 4 and 6
Multiples of 4: 4, 8, 12, 16…
Multiples of 6: 6, 12, 18…

First common multiple = 12 (LCM)

Relationship Between LCM and HCF

For two numbers:

LCM × HCF = Product of the two numbers

This formula is very useful in SSC GD exams.

Methods to Find HCF and LCM

1. Prime Factorization Method

Break each number into prime factors.

• HCF → take common factors with the lowest power
  
  

• LCM → take all factors with the highest power
  
  

Example

Numbers: 72, 96, 144

Prime factors:
72 = 2³ × 3²
96 = 2⁵ × 3
144 = 2⁴ × 3²

HCF = 2³ × 3 = 24
LCM = 2⁵ × 3² = 288

2. Long Division Method (Very Important for SSC GD)

This method is often used when the question gives:

• Last divisor
  
  

• Quotients
  
  

• Difference between numbers
  
  

Standard Formula

Dividend = Divisor × Quotient + Remainder

Example

Last divisor = 49
Quotients = 17, 3, 2

Work backward:

• 49 × 2 = 98
  
  

• 98 × 3 + 49 = 343
  
  

• 343 × 17 + 98 = 5929
  
  

Two numbers = 5929 and 343
Difference = 5586

This type of question is frequently asked in SSC GD.

LCM-Based Word Problems (Most Common)

1. Bell / Alarm Problems

When bells ring at different intervals and you are asked when they will ring together, always use LCM.

Example

Bells ring at intervals of 2, 4, 6, 8, 10, 12 seconds.

LCM = 120 seconds = 2 minutes

If asked:

How many times will they ring together in 30 minutes?

Calculation:
30 ÷ 2 = 15
Since they ring together at the start also, add +1

Answer = 16 times

⚠️ If the question says “after the first time”, do not add +1.

2. Steps (Walking Distance) Problems

When people have different step lengths and the question asks for minimum distance, use LCM.

Example

Step lengths: 36 cm, 24 cm, 48 cm

Minimum distance = LCM = 144 cm

3. Race / Circular Track Problems

When runners complete a track in different times and start together, use LCM.

Example

Times: 30 sec, 40 sec, 50 sec, 60 sec

LCM = 600 seconds = 10 minutes

They will meet again at the same point after 10 minutes.

Direction does not matter in such questions.

HCF-Based Word Problems

1. Classroom / Room Distribution Problems

When asked for:

• Minimum number of rooms
  
  

• Maximum students in each group
  
  

Always use HCF.

Example

Students: 72, 96, 144

HCF = 24

Rooms needed:
72 ÷ 24 = 3
96 ÷ 24 = 4
144 ÷ 24 = 6

Total rooms = 13

2. Remainder-Based Problems

Example

Find the smallest number which leaves remainder 3 when divided by
12, 18, 21, 30

Step 1: Subtract remainder
12−3, 18−3, 21−3, 30−3

Step 2: Find LCM
LCM = 1260

Step 3: Add remainder
Answer = 1263

3. “Double / Half” Type Questions

Example

Find the smallest number which, when doubled, is divisible by
12, 18, 21, 30

LCM = 1260
Required number = 1260 ÷ 2 = 630

Largest Number Problems (HCF Based)

Example

Find the largest number that divides 2300 and 3500 leaving remainders 32 and 56.

Subtract remainders:
2300 − 32 = 2268
3500 − 56 = 3444

HCF of 2268 and 3444 = Answer

SSC GD Exam Tips for LCM and HCF

• “Minimum”, “Together”, “Same time” → LCM

• “Maximum”, “Minimum rooms”, “Largest number” → HCF

• Bell and race problems → usually LCM

• Always read whether +1 is required or not

• Remainder problems → subtract remainder first