HCF and LCM — Study Notes for SSC MTS

Overview

HCF (Highest Common Factor) and LCM (Lowest Common Multiple) form a crucial topic in the SSC MTS quantitative section, appearing in 2–4 questions every year. These questions test both direct calculation skills and problem-solving through word problems involving time intervals, bell ringing, meeting points, and gear rotations.

The topic is foundational because it connects to other areas like fractions, simplification, and rational numbers. Students must master three skill layers: computing HCF/LCM of whole numbers using prime factorization or division methods, extending these concepts to fractions and decimals, and translating real-world scenarios into HCF/LCM problems. Most errors occur in word problems where students confuse when to use HCF versus LCM.

Strong performance here guarantees 6–10 marks and builds confidence for related topics like ratio-proportion and time-work problems that implicitly use these concepts.

Key Concepts

• **HCF (Highest Common Factor)**: The largest number that divides all given numbers without remainder. Used when we need the greatest common measure, largest size of identical groups, or maximum capacity that divides quantities evenly.

• **LCM (Lowest Common Multiple)**: The smallest number that is a multiple of all given numbers. Applied when we need the smallest common interval, first simultaneous occurrence, or minimum quantity divisible by all given numbers.

• **Prime Factorization Method**: Express each number as a product of prime factors. For HCF, take the product of common prime factors with lowest powers. For LCM, take the product of all prime factors with highest powers.

• **Division Method for HCF**: Divide the larger number by the smaller, then divide the previous divisor by the remainder. Continue until remainder is zero; the last divisor is the HCF. This is the Euclidean algorithm.

• **Division Method for LCM**: Arrange numbers in a row, divide by common prime factors, bring down undivided numbers. Continue until no common factors remain. LCM = product of all divisors and remaining numbers.

• **Fundamental Relationship**: For any two numbers a and b: HCF(a,b) × LCM(a,b) = a × b. This formula is heavily tested and helps find one value when three others are known.

• **HCF/LCM of Fractions**: HCF = (HCF of numerators)/(LCM of denominators). LCM = (LCM of numerators)/(HCF of denominators). Remember the criss-cross pattern.

• **HCF/LCM of Decimals**: Convert all decimals to like decimals (same number of decimal places), remove decimal points to get whole numbers, find HCF/LCM of these whole numbers, then place decimal point in the result at appropriate position.

Formulas / Key Facts

**For Whole Numbers:**

• HCF × LCM = Product of two numbers (works only for exactly two numbers)
• HCF of a, b divides both a and b; LCM of a, b is divisible by both a and b
• HCF is always ≤ smallest number; LCM is always ≥ largest number
• HCF of co-prime numbers = 1; LCM of co-prime numbers = their product

**For Fractions:**

• HCF of fractions = (HCF of numerators) / (LCM of denominators)
• LCM of fractions = (LCM of numerators) / (HCF of denominators)

**For Decimals:**

• Convert to same decimal places, then treat as whole numbers
• Place decimal point in result: number of decimal places = maximum among given numbers

**Word Problem Indicators:**

• Use HCF when: "greatest/largest/maximum that divides," "equal groups," "cutting into largest equal pieces"
• Use LCM when: "smallest/least/minimum," "together again," "simultaneous occurrence," "first time"

Worked Examples

**Example 1: Basic HCF and LCM** Find HCF and LCM of 12 and 18.

*Solution:* Prime factorization: 12 = 2² × 3, 18 = 2 × 3² HCF = product of common factors with lowest power = 2¹ × 3¹ = 6 LCM = product of all factors with highest power = 2² × 3² = 36 Verification: 6 × 36 = 216 = 12 × 18 ✓

**Example 2: HCF and LCM of Fractions** Find HCF and LCM of 2/3, 4/9, and 8/27.

*Solution:* Numerators: 2, 4, 8 → HCF = 2, LCM = 8 Denominators: 3, 9, 27 → HCF = 3, LCM = 27 HCF of fractions = 2/27 LCM of fractions = 8/3

**Example 3: Word Problem — Meeting Point** Three bells ring at intervals of 12, 15, and 20 minutes. If they ring together at 9:00 AM, when will they ring together again?

*Solution:* This is an LCM problem (simultaneous occurrence). LCM(12, 15, 20): 12 = 2² × 3, 15 = 3 × 5, 20 = 2² × 5 LCM = 2² × 3 × 5 = 60 minutes = 1 hour They ring together again at 9:00 AM + 1 hour = 10:00 AM

**Example 4: Word Problem — Maximum Size** A rectangular field is 180 m long and 120 m wide. Find the maximum side length of square tiles that can cover the field exactly.

*Solution:* This is an HCF problem (largest measure that divides both). HCF(180, 120): Using division: 180 = 120 × 1 + 60, 120 = 60 × 2 + 0 HCF = 60 m Maximum tile size = 60 m × 60 m

Common Mistakes

**Mistake 1: Confusing HCF and LCM in word problems** Wrong thinking: "Three buses arrive every 6, 8, 10 minutes. When do they arrive together? Must be HCF." Correct fix: "Together" means simultaneous occurrence = LCM. LCM(6,8,10) = 120 minutes.

**Mistake 2: Using HCF × LCM = product formula for more than two numbers** Wrong thinking: For 4, 6, 8: HCF = 2, LCM = 24, so 2 × 24 should equal 4 × 6 × 8. Correct fix: The formula HCF × LCM = product works ONLY for exactly two numbers. For three or more numbers, no such direct relationship exists.

**Mistake 3: Incorrect fraction HCF/LCM formula** Wrong thinking: HCF of fractions = (HCF of numerators)/(HCF of denominators). Correct fix: HCF uses HCF in numerator but LCM in denominator. LCM uses LCM in numerator but HCF in denominator. Remember the criss-cross.

**Mistake 4: Misplacing decimal in HCF/LCM of decimals** Wrong thinking: HCF of 0.6 and 0.9 → treat as 6 and 9 → HCF = 3. Correct fix: 0.6 and 0.9 have one decimal place. HCF(6,9) = 3, now place decimal: 0.3.

**Mistake 5: Forgetting co-prime concept** Wrong thinking: Numbers must have HCF > 1. Correct fix: Co-prime numbers (like 8 and 15) have HCF = 1 and LCM = their product (120). This is valid and commonly tested.

Quick Reference

• **HCF × LCM = Product** (for exactly two numbers only)
• **HCF formula for fractions**: (HCF numerators)/(LCM denominators)
• **LCM formula for fractions**: (LCM numerators)/(HCF denominators)
• **Word problem clue**: "Greatest/maximum that divides" → HCF
• **Word problem clue**: "Least/smallest common" or "together again" → LCM
• **Decimal HCF/LCM**: Match decimal places, compute, restore decimal position