LCM & HCF – Concepts, Prime Factorization & Shortcuts

 

Welcome back to Concept Studies — Because Concepts Matter! As an aspiring competitive exam student, this is one chapter I can confidently say will appear in almost every exam: Factors, Multiples, HCF & LCM. And trust me — if you understand the prime factorization base, this chapter becomes the easiest scoring area in Quantitative Aptitude!

🔹 Why HCF & LCM Are Important?

If you're preparing for SSC, Banking, Railways, MPSC, UPSC Prelims, or any State-level exam, you have already seen these questions:

• Find the LCM of 12, 18, 24
• Three bells ring together… When will they ring together again?
• Find the HCF of 36 and 84 using prime factorization
• A man takes steps of 45 cm and 75 cm. What is the minimum distance both will cover together?

All of these can be solved within 10–20 seconds once your concepts are clear.

🔹 Understanding Factors & Multiples

Factor

A factor is a number that divides another number exactly.

Example: Factors of 12 → 1, 2, 3, 4, 6, 12

Multiple

A multiple is the result of multiplying a number.

Example: Multiples of 4 → 4, 8, 12, 16, 20...

🔹 HCF (Highest Common Factor)

The greatest number that divides all the given numbers exactly.

🔹 LCM (Least Common Multiple)

The smallest number that is divisible by all the given numbers.

🔹 Prime Factorization Method (Universal Method)

This is the most accurate and exam-proof method. All we do is break the numbers into prime factors.

Example:
12 = 2 × 2 × 3  
18 = 2 × 3 × 3

How to find HCF?

Take the common prime factors with minimum powers.

Numbers: 12, 18  
Common prime factors: 2 and 3  
Powers: 2¹ and 3¹  
HCF = 2 × 3 = 6

How to find LCM?

Take all prime factors with maximum powers.

2² × 3² = 4 × 9 = 36  
LCM = 36

🔹 Division Method (Fastest for Two Numbers)

Example:

36 | 12  
    | 4  
        | 1  
HCF = 4

🔹 Shortcut #1 – Relationship Between HCF & LCM

Product of numbers = HCF × LCM

This shortcut is extremely useful in many word problems!

Example:

If HCF = 8 and LCM = 144, product = 8 × 144 = 1152  
If one number = 36  
Other number = 1152 / 36 = 32

🔹 Shortcut #2 – LCM of Fractions

LCM of Fractions = LCM of Numerators / HCF of Denominators

🔹 Shortcut #3 – HCF of Fractions

HCF of Fractions = HCF of Numerators / LCM of Denominators

🔹 Shortcut #4 – LCM of Numbers Close to Each Other

If numbers are consecutive:

LCM of consecutive numbers = Product of numbers

Example: 7, 8, 9 → LCM = 7 × 8 × 9

🔹 Shortcut #5 – HCF of Large Numbers

Subtract the smaller from the larger until numbers become small.

🔹 Common Word Problems

🔸 Type 1: Repetition / Cyclic Events

Example: Three bells ring at 6 sec, 8 sec, 12 sec. When will they ring together?

LCM = 24 sec (Answer)

🔸 Type 2: Minimum Distance / Maximum Length

Example: Find the minimum length of rope that can cut lengths of 20cm, 30cm, 40cm.

HCF = 10cm (Answer)

🔹 Practice Examples (Solve Yourself)

1. Find the HCF and LCM of 15, 25, 35
2. Find the LCM of 3/4, 5/6, 7/8
3. When will 10 min, 15 min and 20 min timers ring together?
4. Two numbers’ LCM = 180 and HCF = 6. One number is 30. Find the other.
5. Find the greatest number that divides 75, 90, 135 exactly.

🔹 Coming Next in Lesson 5

• Fractions & Decimals
• Conversions
• Simplification tricks
• Decimal-based shortcuts

Stay tuned — and remember, Concepts Matter!