LCM & HCF

LCM & HCF — Study Notes for Maharashtra Police Bharti

Overview

LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number-system problems in the Maharashtra Police Bharti exam.

Mastering LCM and HCF requires two skills: quick calculation methods (prime factorisation, division method) and recognising which concept applies to a word problem. The exam tests both computational speed and conceptual clarity, so you must practise until these become second nature.

Key insight: HCF deals with dividing things into smaller equal parts (what's common), while LCM deals with combining cycles or groups (when things align again). Keep this mental model and you'll never confuse which to apply.

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Key Concepts

• HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor).
• LCM (Lowest Common Multiple): The smallest number that is exactly divisible by two or more numbers.
• Fundamental relationship: For any two numbers a and b: LCM × HCF = a × b
• Co-prime numbers: Two numbers with HCF = 1 (e.g., 8 and 15). Their LCM equals their product.
• HCF of fractions: HCF of numerators ÷ LCM of denominators
• LCM of fractions: LCM of numerators ÷ HCF of denominators
• Divisibility rule: HCF always divides LCM. If it doesn't in your calculation, recheck your work.
• For three or more numbers: Find LCM/HCF of first two, then use that result with the third number, and so on.

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Formulas / Key Facts

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Worked Examples

Example 1: Basic Calculation

Find LCM and HCF of 18 and 24.

Step 1: Prime factorisation

• 18 = 2 × 3 × 3 = 2¹ × 3²
• 24 = 2 × 2 × 2 × 3 = 2³ × 3¹

Step 2: For HCF, take lowest powers of common primes

• HCF = 2¹ × 3¹ = 6

Step 3: For LCM, take highest powers of all primes

• LCM = 2³ × 3² = 8 × 9 = 72

Verification: LCM × HCF = 72 × 6 = 432 = 18 × 24 ✓

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Example 2: Word Problem (LCM Application)

Three bells ring at intervals of 4, 6, and 8 minutes. If they ring together at 10:00 AM, when will they ring together again?

Logic: We need the smallest time when all intervals align → LCM

Step 1: Find LCM of 4, 6, 8

• 4 = 2²
• 6 = 2 × 3
• 8 = 2³
• LCM = 2³ × 3 = 24 minutes

Answer: They ring together again at 10:24 AM.

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Example 3: Word Problem (HCF Application)

A room is 15 m long and 12 m wide. Find the largest square tile that can exactly cover the floor.

Logic: We need the largest size that divides both dimensions → HCF

Step 1: Find HCF of 15 and 12

• 15 = 3 × 5
• 12 = 2² × 3
• HCF = 3

Answer: Largest tile size = 3 m × 3 m

Bonus: Number of tiles = (15 × 12) / (3 × 3) = 180 / 9 = 20 tiles

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Example 4: Using the Product Formula

The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.

Formula: LCM × HCF = Product of numbers

• 180 × 12 = 36 × other number
• 2160 = 36 × other number
• Other number = 2160 / 36 = 60

Answer: The other number is 60.

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Example 5: Fractions

Find LCM of 2/3, 4/5, and 6/7.

Step 1: LCM of numerators (2, 4, 6) = 12 Step 2: HCF of denominators (3, 5, 7) = 1 (all co-prime) Answer: LCM = 12/1 = 12

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Common Mistakes

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Quick Reference

• HCF = largest divider → use for cutting, dividing, distributing equally
• LCM = smallest common multiple → use for cycles, bells, meeting points
• Two numbers only: LCM × HCF = a × b
• Prime factorisation: HCF takes minimum powers, LCM takes maximum powers
• Division method shortcut: Divide both numbers by common factors repeatedly; multiply all divisors for HCF
• Sanity check: HCF ≤ smallest number; LCM ≥ largest number

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Quick Problem-Type Identifier