The LCM of 3, 4 and 5 is 60. This comprehensive guide covers everything you need to know about finding the least common multiple of 3, 4 and 5: the pairwise coprime shortcut, prime factorization, division ladder method, listing multiples, step-by-step visual diagrams, real-world applications, and answers to frequently asked questions.
The LCM of 3, 4 and 5 is 60. The Least Common Multiple (LCM) of three numbers is the smallest positive integer that is divisible by all three numbers (3, 4, and 5) without leaving any remainder.
For 3, 4 and 5, the LCM is 60 because:
Visualization of multiples of 3, 4, and 5 on a single number line. While pairs meet at 12 (for 3 and 4), 15 (for 3 and 5), and 20 (for 4 and 5), the very first point where all three lines meet simultaneously is at 60.
There are three standard methods to find the LCM of 3, 4 and 5. Each method provides the exact same result: 60.
Because 3, 4, and 5 share no prime factors pairwise, directly multiply them: 3 × 4 × 5 = 60.
Break 3, 4, and 5 into prime factors, select the highest power of each prime, and multiply.
Divide 3, 4, and 5 together by prime factors until all quotients reach 1, then multiply divisors.
The prime factorization method clearly explains why the coprime shortcut works and is guaranteed to work for all integers.
| Number | Prime Factorization | Prime Factor 2 | Prime Factor 3 | Prime Factor 5 | Highest Power |
|---|---|---|---|---|---|
| 3 | 31 | — | 31 | — | — |
| 4 | 22 | 22 | — | — | — |
| 5 | 51 | — | — | 51 | — |
| LCM | 22 × 3 × 5 | 22 = 4 | 31 = 3 | 51 = 5 | 4 × 3 × 5 = 60 |
Prime factor composition of 3, 4, and 5. Notice that none of the numbers share any common prime factor — their prime factor sets are completely disjoint. Therefore, their LCM is simply the product of their values.
In the common division method (also known as the ladder method), we divide 3, 4, and 5 together by prime factors until all quotients equal 1.
| Prime Divisor | Quotient (from 3) | Quotient (from 4) | Quotient (from 5) | Operation Details |
|---|---|---|---|---|
| 2 | 3 | 2 | 5 | Divide 4 by 2 (bring down 3 and 5) |
| 2 | 3 | 1 | 5 | Divide 2 by 2 (bring down 3 and 5) |
| 3 | 1 | 1 | 5 | Divide 3 by 3 (bring down 1 and 5) |
| 5 | 1 | 1 | 1 | Divide 5 by 5 (all quotients are now 1) |
Division ladder for 3, 4, and 5. Multiply all prime divisors on the left to compute the LCM.
LCM(3, 4, 5) = 2 × 2 × 3 × 5 = 60
List the multiples of 3, 4, and 5 sequentially until you reach the smallest common multiple.
| Number | Multiples | First Common Multiple |
|---|---|---|
| 3 | 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, … | 60 |
| 4 | 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, … | |
| 5 | 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, … |
The LCM and GCD (Greatest Common Divisor) of 3, 4 and 5 represent two foundational complementary mathematical principles. Because 3, 4, and 5 are pairwise coprime, their GCD is 1.
| Aspect | LCM of 3, 4 and 5 | GCD of 3, 4 and 5 |
|---|---|---|
| Definition | Smallest number divisible by 3, 4, and 5 | Largest number that divides 3, 4, and 5 |
| Value | 60 | 1 |
| Relationship | Since 3, 4, 5 are pairwise coprime: LCM = 3 × 4 × 5 = 60, GCD = 1 | |
| Size Rule | Always ≥ 5 (the largest input) | Always ≤ 3 (the smallest input) |
| Primary Use | Adding fractions, synchronizing schedules | Simplifying fractions, grouping items evenly |
The LCM of 3, 4 and 5 frequently appears across science, scheduling, and engineering wherever repeating periods must synchronize.
| Field | Problem | LCM Role | Practical Example |
|---|---|---|---|
| Mathematics | Adding fractions | Find least common denominator | 1/3 + 1/4 + 1/5 = 20/60 + 15/60 + 12/60 = 47/60 |
| Transit / Logistics | Simultaneous departures | Synchronized timing | Buses every 3, 4, 5 min → depart together every 60 min (1 hr) |
| Mechanical Gears | Interlocking gear cycles | Re-alignment rotations | 3, 4, 5-tooth gears return to start after 60 tooth steps |
| Inventory & Packaging | Equal item quantities | Equalization batches | Packets of 3, 4, 5 items → buy 60 of each to equalize |
| Time Division | Clock minute divisions | Clean unit sub-divisions | 60 minutes divides cleanly into 20×3 min, 15×4 min, 12×5 min |
To add 1/3 + 1/4 + 1/5, we calculate the Least Common Denominator (LCD), which is LCM(3, 4, 5) = 60:
If three buses leave from a terminal every 3, 4, and 5 minutes respectively starting at 8:00 AM, they will next leave the terminal together at LCM(3, 4, 5) = 60 minutes later (9:00 AM).
Three solved step-by-step examples demonstrating different approaches.
| Step | Action | Detailed Result |
|---|---|---|
| 1 | Check GCD of each pair | GCD(3,4) = 1; GCD(3,5) = 1; GCD(4,5) = 1 (pairwise coprime) |
| 2 | Apply coprime rule: LCM = a × b × c | LCM = 3 × 4 × 5 = 60 |
| ✓ | Verify divisibility | 60 ÷ 3 = 20 ✓; 60 ÷ 4 = 15 ✓; 60 ÷ 5 = 12 ✓ |
Find LCM(3, 4, 5) using prime factor decomposition:
Find the first common multiple among the multiples of 3, 4, and 5:
Avoid these common misconceptions when finding the LCM of 3, 4 and 5.
| ❌ Mistake | Why it's wrong | ✅ Correct approach |
|---|---|---|
| Saying the LCM is 120 | 120 is a common multiple, but not the least common multiple | The LCM of 3, 4 and 5 is 60 |
| Confusing LCM with GCD | GCD(3, 4, 5) = 1, but LCM(3, 4, 5) = 60 | Remember: LCM ≥ max(3, 4, 5); GCD ≤ min(3, 4, 5) |
| Assuming LCM = product always | Only works when numbers are pairwise coprime. For 4 and 6, this fails | Always check pairwise GCDs first — here it works because 3, 4, 5 are pairwise coprime |
| Using LCM × GCD = product for 3 numbers | The formula LCM × GCD = product is only valid for exactly two numbers | For three pairwise coprime numbers, LCM = product directly |
Answers to the most frequently asked questions regarding the LCM of 3, 4 and 5.
The LCM of 3, 4 and 5 is 60. It is the smallest positive integer that is divisible by all three numbers: 3, 4, and 5 without leaving any remainder.
You can find the LCM of 3, 4 and 5 using three methods: (1) Coprime Shortcut — since 3, 4, and 5 are pairwise coprime, LCM = 3 × 4 × 5 = 60; (2) Prime Factorization — 3 = 3, 4 = 2², 5 = 5, so LCM = 2² × 3 × 5 = 60; or (3) Listing Multiples — list multiples until you find 60.
The LCM of 3, 4, 5, and 6 is 60. Since 60 is divisible by 6 (60 ÷ 6 = 10), adding 6 does not increase the LCM.
No. While 120 is a common multiple of 3, 4 and 5, it is not the least common multiple. The LCM of 3, 4 and 5 is 60. 120 is simply the second common multiple (60 × 2).
The GCD (Greatest Common Divisor) of 3, 4 and 5 is 1. Since they share no common factor other than 1, they are pairwise coprime.
Since 3, 4, and 5 are pairwise coprime (GCD of every pair = 1), their LCM equals their product: 3 × 4 × 5 = 60. Also, LCM(3,4,5) × GCD(3,4,5) = 60 × 1 = 60, which happens to equal the product.
LCM(3, 4, 5) = 60 is used when adding fractions with denominators 3, 4, and 5, when scheduling events that repeat every 3, 4, and 5 minutes, and when finding when three gears with 3, 4, and 5 teeth will realign.
Yes. Three numbers are pairwise coprime if the GCD of every pair is 1. For 3, 4, and 5: GCD(3,4) = 1, GCD(3,5) = 1, and GCD(4,5) = 1. This is why LCM(3, 4, 5) = 3 × 4 × 5 = 60.
Key terms related to the LCM of 3, 4 and 5.