The LCM of 6, 8, and 10 is 120. All three numbers share a factor of 2, and their prime factor breakdown requires finding the highest powers of 2, 3, and 5. This guide walks through everything — three methods, a custom 3-circle Venn diagram visualization, worked examples, the GCD relationship, real-world applications, and FAQs.
The LCM of 6, 8, and 10 is 120. The Least Common Multiple (LCM) of a set of numbers is the smallest positive integer that is divisible by all of them without leaving a remainder.
For 6, 8, and 10, the LCM is 120 because:
Visualization of the first 15 multiples of 6, 8, and 10 on a number line using the brand color palette. The first common multiple across all three is 120.
There are three standard methods to find the LCM of 6, 8, and 10. Each one gives the exact same result: 120.
List the multiples of 6, 8, and 10 until you find the smallest number they all share.
Break 6, 8, and 10 into prime factors, take the highest power of each prime, and multiply.
Divide 6, 8, and 10 by common prime factors until no further division is possible.
Prime factorization is the most reliable method — it breaks numbers down into their prime components.
Designed with website theme colors: The center intersection holds the common prime factor 2 (GCD = 2). The unique parts hold 3 (from 6), two 2s (from 8), and 5 (from 10). The union of all regions gives 3 × 2 × (2 × 2) × 5 = 2³ × 3 × 5 = 120 (LCM).
| Number | Prime Factorization | 2 | 3 | 5 | Highest Power |
|---|---|---|---|---|---|
| 6 | 2 × 3 | 21 | 31 | — | — |
| 8 | 23 | 23 | — | — | — |
| 10 | 2 × 5 | 21 | — | 51 | — |
| LCM | 23 × 3 × 5 | 23 | 31 | 51 | 8 × 3 × 5 = 120 |
In the ladder method, we divide all three numbers by common primes until no prime divides at least two numbers.
| Prime Factor | 6 | 8 | 10 |
|---|---|---|---|
| 2 | 3 | 4 | 5 |
| 2 | 3 | 2 | 5 |
| 2 | 3 | 1 | 5 |
| 3 | 1 | 1 | 5 |
| 5 | 1 | 1 | 1 |
LCM(6, 8, 10) = 2 × 2 × 2 × 3 × 5 = 120
List the multiples of 6, 8, and 10 until you find the smallest common multiple across all three lists.
| Number | Multiples | First Common |
|---|---|---|
| 6 | 6, 12, 18, 24, 30, 36, 42, 48, …, 120 | 120 |
| 8 | 8, 16, 24, 32, 40, 48, 56, …, 120 | |
| 10 | 10, 20, 30, 40, 50, 60, 70, …, 120 |
The LCM and GCD (Greatest Common Divisor) of 6, 8, and 10 are different values that serve complementary purposes in mathematics.
| Aspect | LCM of 6, 8, 10 | GCD of 6, 8, 10 |
|---|---|---|
| Definition | Smallest number divisible by 6, 8, and 10 | Largest number that divides 6, 8, and 10 |
| Value | 120 | 2 |
| Relationship | LCM × GCD identity only applies pairwise (for 2 numbers at a time) | |
| Size | Always ≥ 10 (the largest input) | Always ≤ 6 (the smallest input) |
| Primary Use | Adding fractions, event schedules | Grouping items, simplifying fractions |
The LCM of 6, 8, and 10 appears in many practical situations where repeating events need to align.
| Field | Problem | LCM Role | Example |
|---|---|---|---|
| Mathematics | Adding fractions | Find common denominator | 1/6 + 1/8 + 1/10 = 20/120 + 15/120 + 12/120 = 47/120 |
| Scheduling | Repeating events | When events align | Bus every 6, 8 & 10 min → together every 120 min (2 hours) |
| Mechanics | Gear rotation | Re-alignment point | 6, 8 & 10-tooth gears → realign after 120 turns (20, 15, and 12 rotations) |
| Commerce | Package sizes | Equal quantities | 6-pack, 8-pack & 10-pack → buy 120 of each (20, 15, and 12 packs) |
| Music | Rhythm patterns | Pattern re-sync | 6/8, 8/8 & 10/8 time phrases → realign every 120 beats |
To add 1/6 + 1/8 + 1/10, find the LCD (Least Common Denominator), which is LCM(6, 8, 10) = 120.
If one bus leaves every 6 minutes, another every 8 minutes, and a third every 10 minutes, they will all depart together every LCM(6, 8, 10) = 120 minutes (exactly 2 hours).
Three fully solved examples using different methods.
| Step | Action | Result |
|---|---|---|
| 1 | Prime factorize 6, 8, 10 | 6 = 2 × 3; 8 = 2³; 10 = 2 × 5 |
| 2 | Take highest powers | 2³, 3¹, 5¹ |
| 3 | Multiply | 8 × 3 × 5 = 120 |
| ✓ | Verify | 120 ÷ 6 = 20 ✓; 120 ÷ 8 = 15 ✓; 120 ÷ 10 = 12 ✓ |
You can also find LCM of three numbers by folding pairwise:
Find the first common multiple of 6, 8, and 10.
Here are common LCM values for sets that frequently appear alongside LCM(6, 8, 10).
Here are the most common errors made on this set — and how to avoid them.
| ❌ Mistake | Why it's wrong | ✅ Correct approach |
|---|---|---|
| Saying the LCM is 480 | 480 is the product 6 × 8 × 10, but not the least common multiple | The LCM of 6, 8, and 10 is 120 |
| Multiplying 6 × 8 × 10 = 480 | Only works when all numbers are pairwise coprime (which they aren't here) | Use prime factorization: LCM = 2³ × 3 × 5 = 120 |
| Saying the LCM is 24 (using only 6 and 8) | 24 is not divisible by 10 | Include all three numbers: LCM = 120 |
| Taking 2¹ instead of 2³ | 8 has 2³, and the highest power wins | Use 2³ = 8 for the prime 2 |
Quick answers to the most common questions about the LCM of 6, 8, and 10.
The LCM of 6, 8, and 10 is 120. It is the smallest positive integer that is divisible by all three numbers.
Use prime factorization: 6 = 2 × 3, 8 = 2³, and 10 = 2 × 5. Take the highest power of each prime: 2³ × 3 × 5 = 8 × 15 = 120. You can also use the Venn diagram, division ladder method, or list multiples until you reach 120.
The GCD (Greatest Common Divisor) of 6, 8, and 10 is 2. All three numbers are even, and 2 is the largest number that divides all three evenly.
In a 3-circle Venn diagram, the central 3-way overlap has the common factor 2 (GCD = 2). The unique regions contain 3 (from 6), two 2s (from 8), and 5 (from 10). Multiplying all regions across the diagram gives 3 × 2 × (2 × 2) × 5 = 120 (LCM).
The LCM of 6, 8, 10, and 12 is 120. Since 120 is already divisible by all four numbers, adding 12 to the list doesn't change the LCM.
No. 480 is the product 6 × 8 × 10, but it is not the least common multiple. Because 6, 8, and 10 share factors, their actual LCM is 120 — smaller than their product.
The LCM of 6 and 8 is 24. But 24 is not divisible by 10, which is why we need a larger LCM when we include 10.
LCM(6, 8, 10) = 120 is used when adding fractions with denominators 6, 8, and 10; when scheduling events that repeat every 6, 8, and 10 minutes; and when aligning gears with 6, 8, and 10 teeth.
Key terms related to the LCM of 6, 8, and 10.