LCM and GCD: Complete Guide with Examples

LCM (Least Common Multiple) and GCD (Greatest Common Divisor) are fundamental concepts in mathematics used in fractions, ratios, time calculations, and real-world problem-solving. This guide explains both concepts with multiple calculation methods and practical examples.

What is GCD (Greatest Common Divisor)?

The GCD (also called HCF - Highest Common Factor) is the largest number that divides all given numbers without leaving a remainder.

What is LCM (Least Common Multiple)?

The LCM is the smallest number that all given numbers divide into evenly.

Method 1: Prime Factorization (Best for Understanding)

Finding GCD Using Prime Factorization

Finding LCM Using Prime Factorization

Method 2: Euclidean Algorithm (Fastest for GCD)

The Euclidean algorithm finds GCD by repeatedly dividing and taking remainders.

Steps:

1. Divide the larger number by the smaller
2. Replace the larger with the smaller, and the smaller with the remainder
3. Repeat until remainder is 0
4. The last non-zero remainder is the GCD

Method 3: Using the GCD-LCM Relationship

There's a useful relationship between GCD and LCM:

LCM(a, b) = (a × b) ÷ GCD(a, b)

Real-World Applications

1. Scheduling Problems (LCM)

Problem: Bus A arrives every 12 minutes, Bus B every 18 minutes. When do they arrive together?

Solution: LCM(12, 18) = 36 minutes

They arrive together every 36 minutes.

2. Cutting Materials into Equal Parts (GCD)

Problem: You have wooden planks of 48 cm and 60 cm. What's the longest piece you can cut both into without waste?

Solution: GCD(48, 60) = 12 cm

Cut both into 12 cm pieces.

3. Simplifying Fractions (GCD)

Problem: Simplify 48/60

Solution: GCD(48, 60) = 12

48 ÷ 12 = 4, 60 ÷ 12 = 5

Simplified: 4/5

4. Adding Fractions (LCM)

Problem: Add 1/12 + 1/18

Solution: LCM(12, 18) = 36 (common denominator)

= 3/36 + 2/36 = 5/36

LCM and GCD for More Than 2 Numbers

Example: GCD of 24, 36, 48

24 = 2³ × 3¹

36 = 2² × 3²

48 = 2⁴ × 3¹

GCD = 2² × 3¹ = 4 × 3 = 12

Example: LCM of 24, 36, 48

LCM = 2⁴ × 3² = 16 × 9 = 144

Common Mistakes to Avoid

Mistake 1: Confusing GCD and LCM

Remember: GCD is always ≤ smallest number. LCM is always ≥ largest number.

Mistake 2: Missing Prime Factors

For LCM, include ALL prime factors from all numbers, not just common ones.

Mistake 3: Wrong Powers in Prime Factorization

GCD uses LOWEST powers, LCM uses HIGHEST powers.

Quick Reference Table

Practice Problems

1. Find GCD and LCM of 24 and 36
2. Two gears with 40 and 60 teeth. After how many rotations do they sync?
3. Simplify the fraction 72/96 using GCD
4. Add 1/8 + 1/12 using LCM for the denominator