LCM and HCF — Study Notes

Overview

LCM (Least Common Multiple) and HCF (Highest Common Factor, also called GCD — Greatest Common Divisor) form a fundamental pillar of Railway Group D Mathematics. These concepts appear not just as direct calculation problems but also disguised in word problems involving bells ringing together, meeting points of runners, grouping items, and more. Mastery of LCM and HCF is non-negotiable because they underpin questions on simplification, fractions, ratio-proportion, and time-based scenarios.

Railway Group D typically asks 2–3 questions from this topic. You will encounter three types: (1) straightforward calculation of LCM/HCF of whole numbers, (2) LCM/HCF of fractions and decimals, and (3) real-world word problems. The key to scoring full marks is speed and accuracy in prime factorization, understanding the relationship between LCM and HCF, and recognizing when a problem requires which concept.

The good news: this topic is entirely formula-driven and practice-oriented. Once you internalize the methods and practice 20–30 varied problems, you can solve any question in under a minute.

Key Concepts

• **HCF (Highest Common Factor)** is the largest number that divides all given numbers without leaving a remainder. It represents the "biggest common piece" you can extract from the numbers. Use HCF when the problem asks for maximum size, greatest length, or largest group.

• **LCM (Least Common Multiple)** is the smallest number that is exactly divisible by all given numbers. It represents the first common meeting point. Use LCM when the problem asks for minimum time, shortest distance, or when events repeat together.

• **Prime Factorization Method**: Break each number into prime factors. For HCF, take the **lowest power** of common primes. For LCM, take the **highest power** of all primes present.

• **Division Method (for HCF)**: Use Euclidean algorithm — repeatedly divide the larger number by the smaller, replace the larger with the remainder, until remainder is zero. The last non-zero divisor is the HCF.

• **Relation between LCM and HCF**: For any two numbers a and b, **Product of numbers = LCM × HCF**, i.e., a × b = LCM(a,b) × HCF(a,b). This is one of the most tested formulas in the exam.

• **LCM and HCF of fractions**: HCF of fractions = (HCF of numerators) / (LCM of denominators). LCM of fractions = (LCM of numerators) / (HCF of denominators).

• **LCM and HCF of decimals**: Convert decimals to fractions (or multiply by 10, 100, etc. to remove decimal), find LCM/HCF, then convert back by dividing by the same power of 10.

• **Co-prime numbers**: Two numbers whose HCF is 1. For co-prime numbers, LCM = product of the numbers.

Formulas / Key Facts

1. **Product formula**: a × b = LCM(a, b) × HCF(a, b) 2. **HCF of fractions** = (HCF of numerators) / (LCM of denominators) 3. **LCM of fractions** = (LCM of numerators) / (HCF of denominators) 4. **HCF of decimals**: Multiply all numbers by the same power of 10 to make them whole, find HCF, then divide result by that power of 10. 5. **LCM of decimals**: Same conversion method, find LCM, then divide back. 6. **For three or more numbers**: Apply LCM/HCF pairwise or use prime factorization for all at once. 7. **HCF is always ≤ smallest number**; LCM is always ≥ largest number. 8. **Bell/event problems**: Use LCM to find when events coincide again. 9. **Grouping/division problems**: Use HCF to find maximum equal groups or maximum size per group. 10. **Runner/circular track problems**: LCM gives meeting time; HCF rarely used here.

Worked Examples

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

*Prime Factorization:*

• 12 = 2² × 3
• 18 = 2 × 3²

*HCF* = product of lowest powers of common primes = 2¹ × 3¹ = 6 *LCM* = product of highest powers of all primes = 2² × 3² = 36

*Verification:* 12 × 18 = 216; LCM × HCF = 36 × 6 = 216 ✓

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**Example 2: Find LCM and HCF of 2/3, 4/9, 5/6**

*Numerators:* 2, 4, 5 HCF(2, 4, 5) = 1 LCM(2, 4, 5) = 20

*Denominators:* 3, 9, 6 HCF(3, 9, 6) = 3 LCM(3, 9, 6) = 18

*HCF of fractions* = 1/18 *LCM of fractions* = 20/3

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**Example 3: Three bells ring at intervals of 6, 8, and 12 minutes. If they ring together at 9:00 AM, when will they ring together again?**

We need the LCM of 6, 8, 12.

Prime factorization:

• 6 = 2 × 3
• 8 = 2³
• 12 = 2² × 3

LCM = 2³ × 3 = 24 minutes

They will ring together again at 9:00 AM + 24 minutes = **9:24 AM**.

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**Example 4: Find the greatest length that can measure 225 cm, 315 cm, and 360 cm exactly.**

We need HCF of 225, 315, 360.

Prime factorization:

• 225 = 3² × 5²
• 315 = 3² × 5 × 7
• 360 = 2³ × 3² × 5

Common primes with lowest powers: 3² × 5 = 45

**Greatest length = 45 cm**.

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**Example 5: Two numbers are in ratio 3:4 and their LCM is 180. Find the numbers.**

Let numbers be 3x and 4x (co-prime multipliers). Since 3 and 4 are co-prime, HCF(3x, 4x) = x.

Using product formula: 3x × 4x = LCM × HCF 12x² = 180 × x 12x = 180 x = 15

Numbers are 3×15 = **45** and 4×15 = **60**.

Common Mistakes

1. **Mixing up LCM and HCF**: Students use LCM when HCF is needed and vice versa. **Fix**: LCM for "together again" or "minimum"; HCF for "maximum equal parts" or "largest measure."

2. **Incorrectly applying fraction formulas**: Writing LCM of fractions as (LCM of numerators)/(LCM of denominators). **Fix**: Remember — LCM fraction uses HCF in denominator; HCF fraction uses LCM in denominator. They swap.

3. **Forgetting to convert decimals**: Attempting to find LCM of 0.6 and 0.8 directly without converting. **Fix**: Multiply by 10 to get 6 and 8, find LCM = 24, divide by 10 to get 2.4.

4. **Using product formula for three or more numbers**: a × b × c ≠ LCM × HCF for three numbers. **Fix**: The product formula works **only for two numbers**. For more, use prime factorization.

5. **Skipping verification in word problems**: Calculating LCM/HCF but not checking if the answer makes sense in context. **Fix**: Always plug your answer back into the problem statement to verify logic (e.g., does 45 cm actually divide all three lengths?).

Quick Reference

• **LCM** = smallest common multiple; use for "together again" problems.
• **HCF** = largest common factor; use for "maximum size/group" problems.
• **Product rule** (two numbers only): a × b = LCM × HCF.
• **LCM of fractions** = (LCM numerators)/(HCF denominators).
• **HCF of fractions** = (HCF numerators)/(LCM denominators).
• **Decimals**: Multiply by 10^n, calculate, divide result by 10^n.
• **Prime factorization**: HCF takes lowest powers of common primes; LCM takes highest powers of all primes.
• **Co-prime numbers**: HCF = 1, so LCM = product.