LCM Made Simple: Complete Guide to Least Common Multiple

The Least Common Multiple (LCM) is the flip side of the GCF—instead of finding the largest number that divides into multiple numbers, you're finding the smallest number that all numbers divide into. This concept is essential for adding fractions, scheduling repeating events, and solving countless real-world problems. This complete guide shows you three proven methodswith clear examples you can practice using MathAI GPT.

1) What is the Least Common Multiple?

The LCM of two or more numbers is the smallest positive number that is a multiple of all of them. Think of it as the first time all the numbers “meet up” when counting by each one.

Example: LCM of 4 and 6 is 12 (the smallest number divisible by both 4 and 6)
Example: LCM of 3 and 5 is 15
Example: LCM of 2, 3, and 4 is 12

LCM vs. GCF: What's the Difference?

GCF (Greatest Common Factor): Largest number that divides into both numbers → “breaking down”

Example: GCF(12, 18) = 6

LCM (Least Common Multiple): Smallest number that both divide into → “building up”

Example: LCM(12, 18) = 36

Use GCF for: Simplifying fractions, dividing into equal groups

Use LCM for: Adding fractions, finding when events coincide

2) Method 1: Listing Multiples

The most intuitive method—list multiples of each number until you find the first one they share.

Worked Example: LCM(4, 6)

Step 1: List multiples of each number

Multiples of 4: 4, 8, 12, 16, 20, 24...

Multiples of 6: 6, 12, 18, 24, 30...

Step 2: Find the smallest common multiple

Result: LCM = 12

Pros & Cons

✅ Pros: Simple, visual, great for small numbers

❌ Cons: Time-consuming for large numbers or multiple numbers

Best for: Numbers under 20, teaching the concept

3) Method 2: Prime Factorization

The most reliable method for any size numbers:

Find the prime factorization of each number
Identify all prime factors that appear
For each prime, take the highest power that appears
Multiply these together

Worked Example: LCM(12, 18)

Step 1: Prime factorization

12 = 2² × 3¹

18 = 2¹ × 3²

Step 2: List all primes that appear: 2 and 3

Step 3: Take highest powers:

Highest power of 2: 2² (from 12)

Highest power of 3: 3² (from 18)

Step 4: Multiply: 2² × 3² = 4 × 9 = 36

Why highest power? We need a number that contains at least as many of each prime as any of the originals

Quick Tip: Compare with GCF

GCF: Take lowest power of common primes only

LCM: Take highest power of all primes that appear

For 12 (2² × 3) and 18 (2 × 3²):

GCF = 2¹ × 3¹ = 6 (lowest common powers)

LCM = 2² × 3² = 36 (highest all powers)

4) Method 3: Using the GCF Shortcut

Here's a powerful relationship between LCM and GCF:

The LCM-GCF Formula

LCM(a, b) × GCF(a, b) = a × b

Rearranged:

LCM(a, b) = (a × b) / GCF(a, b)

When to use: If you already know the GCF, this is the fastest method!

Worked Example: LCM(24, 36)

Step 1: Find GCF(24, 36) = 12

Step 2: Apply formula

LCM = (24 × 36) / 12

LCM = 864 / 12

LCM = 72

Verify: 72 ÷ 24 = 3 ✓ and 72 ÷ 36 = 2 ✓

5) LCM with More Than Two Numbers

To find LCM of three or more numbers:

Example: LCM(4, 6, 8)

Method 1 (Sequential): Find LCM of first two, then LCM of that result with the third

LCM(4, 6) = 12

LCM(12, 8) = 24

Result: 24

Method 2 (Prime Factorization):

4 = 2²

6 = 2 × 3

8 = 2³

Highest powers: 2³ × 3¹ = 8 × 3 = 24

6) Real-World Applications

Adding and Subtracting Fractions

Example: Add 1/4 + 1/6

Problem: You need a common denominator

Step 1: Find LCM(4, 6) = 12

Step 2: Convert fractions

1/4 = 3/12 (multiply by 3/3)

1/6 = 2/12 (multiply by 2/2)

Step 3: Add: 3/12 + 2/12 = 5/12

The LCM gives you the least common denominator (LCD)—the smallest denominator that works!

Scheduling Repeating Events

Example: When Will They Meet Again?

Problem: Bus A arrives every 12 minutes, Bus B every 18 minutes. If both arrive now, when will they arrive together again?

Solution: LCM(12, 18) = 36 minutes

They'll both arrive together in 36 minutes, then again every 36 minutes after that

Work Schedules and Rotations

Example: Work Shift Overlap

Problem: Three employees work every 3, 4, and 5 days respectively. If they all work today, when will they all work together again?

Solution: LCM(3, 4, 5)

3 = 3, 4 = 2², 5 = 5

LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60 days

Shopping: Package Sizes

Example: Buying Exact Quantities

Problem: Hot dogs come in packs of 8, buns in packs of 12. How many of each should you buy to have equal numbers?

Solution: LCM(8, 12) = 24

Buy 3 packs of hot dogs (3 × 8 = 24)

Buy 2 packs of buns (2 × 12 = 24)

Classic problem! The LCM tells you the smallest purchase that works

7) Special Cases

When Numbers Share No Common Factors

If GCF = 1 (coprime numbers): LCM = a × b

Example: LCM(7, 11)

Since 7 and 11 are both prime (GCF = 1)

LCM = 7 × 11 = 77

When numbers are coprime, just multiply them!

When One Number Divides the Other

If a divides b: LCM(a, b) = b (the larger number)

Example: LCM(6, 18)

Since 18 is a multiple of 6

LCM = 18

The larger number already contains the smaller one!

8) Common Mistakes to Avoid

Confusing LCM with GCF: LCM is always ≥ the largest number; GCF is always ≤ the smallest number
Using lowest power instead of highest: For LCM, always take the highest power of each prime
Forgetting primes that appear in only one number: Include ALL primes, even if they only appear once
Multiplying when unnecessary: If one number divides the other, the LCM is just the larger number
Sequential LCM errors: When finding LCM of 3+ numbers sequentially, make sure to continue with all numbers

9) Practice Problems

Find LCM(6, 8) using any method
Find LCM(15, 20) using prime factorization
Find LCM(9, 12, 15)
If GCF(24, 36) = 12, find LCM(24, 36)
What's the LCD (least common denominator) for adding 2/9 + 5/12?
Two gears rotate: one completes a rotation every 8 seconds, the other every 12 seconds. When do they align again?
Find LCM(7, 13) (hint: both are prime)
Find LCM(10, 25)

Show answers

1. Multiples: 6,12,18,24... and 8,16,24... → LCM = 24

2. 15 = 3 × 5; 20 = 2² × 5; LCM = 2² × 3 × 5 = 60

3. 9 = 3²; 12 = 2² × 3; 15 = 3 × 5; LCM = 2² × 3² × 5 = 180

4. LCM = (24 × 36) / 12 = 864 / 12 = 72

5. LCM(9, 12) = 36, so LCD = 36

6. LCM(8, 12) = 24 seconds

7. Both prime, so LCM = 7 × 13 = 91

8. 25 contains 10? No. 10 = 2 × 5; 25 = 5²; LCM = 2 × 5² = 50

Quick Reference Guide

Method 1: List multiples, find first common one

Method 2: Prime factorization, take highest powers of all primes

Method 3: LCM = (a × b) / GCF(a, b)

If coprime: LCM = a × b

If a divides b: LCM = b

For fractions: LCM of denominators = LCD

For scheduling: LCM = when events coincide again

For quick calculations, use an LCM calculator. Need both LCM and GCF? Try the combined LCM & GCF calculator.

Next Steps

Understanding LCM opens the door to working confidently with fractions, solving scheduling problems, and seeing patterns in numbers. It's the perfect complement to GCF—together they give you complete control over factors and multiples.

Want more practice with LCM? Visit MathAI GPT and paste in any LCM problem. Get step-by-step solutions with clear explanations, plus personalized practice problems. Whether you're adding fractions or solving real-world scheduling puzzles, we'll help you master it!

Related topic: Already understand GCF? Check out our complete GCF guide and see how GCF and LCM work together to solve even more problems!