The LCM of 2, 3, and 5 is 30. Since all three numbers are prime and pairwise coprime, their LCM is simply their direct product. This comprehensive guide walks through everything — three standard methods, a custom 3-circle Venn diagram, the coprime shortcut, worked examples, the GCD relationship, real-world applications, and FAQs.
The LCM of 2, 3, and 5 is 30. 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 2, 3, and 5, the LCM is 30 because:
Visualization of the multiples of 2, 3, and 5 on a number line using the brand color palette. All three sequences align at 30.
The fastest way to find LCM(2, 3, 5) is to use the coprime shortcut. Since 2, 3, and 5 are all prime — and therefore pairwise coprime — their LCM is simply their product.
Let's verify. The factors of 2 are 1, 2, the factors of 3 are 1, 3, and the factors of 5 are 1, 5. The only common factor across all three is 1 — so 2, 3, and 5 are pairwise coprime. That means:
This shortcut works for any set of numbers where every pair is coprime. Common cases include:
There are three standard methods to find the LCM of 2, 3, and 5. All three give the exact same result: 30.
List the multiples of 2, 3, and 5 until you find the smallest number they all share.
Break 2, 3, and 5 into prime factors, take the highest power of each prime, and multiply.
Since 2, 3, and 5 are all prime and pairwise coprime, just multiply: 2 × 3 × 5 = 30.
Since 2, 3, and 5 are all prime numbers, the prime factorization is straightforward:
Designed with website theme colors: Since 2, 3, and 5 are pairwise coprime, there are no shared prime factors in any overlaps (Central GCD = 1). The union of all individual circles gives 2 × 3 × 5 = 30 (LCM).
| Number | Prime Factorization | 2 | 3 | 5 | Highest Power |
|---|---|---|---|---|---|
| 2 | 21 | 21 | — | — | — |
| 3 | 31 | — | 31 | — | — |
| 5 | 51 | — | — | 51 | — |
| LCM | 2 × 3 × 5 | 21 | 31 | 51 | 2 × 3 × 5 = 30 |
In the ladder method, we divide all three numbers by common primes until each reaches 1.
| Prime Divisor | 2 | 3 | 5 |
|---|---|---|---|
| 2 | 1 | 3 | 5 |
| 3 | 1 | 1 | 5 |
| 5 | 1 | 1 | 1 |
LCM(2, 3, 5) = 2 × 3 × 5 = 30
List the multiples of 2, 3, and 5 until you find the very first common multiple across all three lists.
| Number | Multiples | First Common |
|---|---|---|
| 2 | 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, … | 30 |
| 3 | 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, … | |
| 5 | 5, 10, 15, 20, 25, 30, 35, 40, … |
The LCM and GCD (Greatest Common Divisor) of 2, 3, and 5 serve complementary purposes in mathematics. Since 2, 3, and 5 are pairwise coprime, their GCD is 1.
| Aspect | LCM of 2, 3, 5 | GCD of 2, 3, 5 |
|---|---|---|
| Definition | Smallest number divisible by 2, 3, and 5 | Largest number that divides 2, 3, and 5 |
| Value | 30 | 1 |
| Pairwise Property | GCD(2, 3) = GCD(2, 5) = GCD(3, 5) = 1 → Pairwise Coprime | |
| Size | Always ≥ 5 (the largest input) | Always ≤ 2 (the smallest input) |
| Primary Use | Adding fractions, scheduling periodic events | Grouping, simplifying fractions |
The LCM of 2, 3, and 5 appears frequently in daily life situations where repeating cycles need to synchronize.
| Field | Problem | LCM Role | Example |
|---|---|---|---|
| Mathematics | Adding fractions | Find common denominator | 1/2 + 1/3 + 1/5 = 15/30 + 10/30 + 6/30 = 31/30 |
| Scheduling | Repeating events | When events align | Buses every 2, 3 & 5 min → together every 30 min |
| Mechanics | Gear rotation | Re-alignment point | 2, 3 & 5-tooth gears → realign after 30 teeth revolutions |
| Commerce | Package sizes | Equal quantities | 2-pack, 3-pack & 5-pack → buy 30 of each (15, 10, and 6 packs) |
| Music | Rhythm patterns | Pattern synchronization | 2/8, 3/8 & 5/8 time phrases → realign every 30 beats |
To compute 1/2 + 1/3 + 1/5, find the LCD (Least Common Denominator), which is LCM(2, 3, 5) = 30:
If three buses leave every 2 minutes, 3 minutes, and 5 minutes respectively, they will all depart together every LCM(2, 3, 5) = 30 minutes.
Step-by-step verification using different mathematical approaches.
| Step | Action | Result |
|---|---|---|
| 1 | Check GCDs of all pairs | GCD(2, 3) = GCD(2, 5) = GCD(3, 5) = 1 |
| 2 | All pairwise coprime → apply rule | LCM = 2 × 3 × 5 = 30 |
| ✓ | Verify | 30 ÷ 2 = 15 ✓; 30 ÷ 3 = 10 ✓; 30 ÷ 5 = 6 ✓ |
You can also find the LCM of three numbers by folding pairwise:
Find the first common multiple of 2, 3, and 5.
Here are common LCM values for sets that frequently appear alongside LCM(2, 3, 5).
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 10 | 10 is only divisible by 2 and 5 — not by 3 | The LCM must be divisible by ALL three: LCM = 30 |
| Adding the numbers: 2 + 3 + 5 = 10 | Addition doesn't give the LCM | Multiply (since they're pairwise coprime): 2 × 3 × 5 = 30 |
| Saying LCM = 6 (just using 2 and 3) | 6 is not divisible by 5 | Include all three numbers: LCM = 30 |
| Applying the shortcut to (2, 4, 5) | 2 and 4 share a factor of 2 — not pairwise coprime | Use ladder or prime factorization for (2, 4, 5): LCM = 20 |
Answers to common questions about the LCM of 2, 3, and 5.
The LCM of 2, 3, and 5 is 30. Since 2, 3, and 5 are all prime and pairwise coprime, their LCM is simply their product: 2 × 3 × 5 = 30.
The fastest way is the coprime shortcut: since GCD(2, 3) = GCD(2, 5) = GCD(3, 5) = 1, LCM(2, 3, 5) = 2 × 3 × 5 = 30. You can also use a 3-set Venn diagram, prime factorization, the ladder method, or list multiples until you reach 30.
The GCD (Greatest Common Divisor) of 2, 3, and 5 is 1. Since they share no common factor other than 1, they are called pairwise coprime.
The LCM of 2, 3, 5, and 7 is 210. Because they are all prime and pairwise coprime, LCM = 2 × 3 × 5 × 7 = 210.
Yes — they are pairwise coprime. Every pair (2, 3), (2, 5), and (3, 5) has GCD = 1. That's why their LCM is simply their direct product.
In a 3-circle Venn diagram for prime factors of 2, 3, and 5, each prime is distinct with no shared factors in the two-way or three-way intersections. The central intersection is 1 (GCD = 1), and multiplying the disjoint regions gives 2 × 3 × 5 = 30 (LCM).
LCM(2, 3, 5) = 30 is used when adding fractions with denominators 2, 3, and 5; when scheduling events that repeat every 2, 3, and 5 minutes; and when aligning gears with 2, 3, and 5 teeth.
Yes. 30 is the smallest positive integer divisible by all of 2, 3, and 5 without leaving a remainder.
Key terms related to the LCM of 2, 3, and 5.