Least Common Multiple Calculator Find the LCM of integers for scheduling, fractions, and cycles — with prime factors, GCF...
Least Common Multiple Calculator
Find the LCM of integers for scheduling, fractions, and cycles — with prime factors, GCF links, and real-world insights.
LCM via GCF (fastest)**:
$$ \text{LCM}(a, b) = \frac{|a \cdot b|}{\gcd(a, b)} $$
Prime Factorization**:
12 = 2² × 3¹ 18 = 2¹ × 3² → LCM = 2max(2,1) × 3max(1,2) = 2² × 3² = **36**
Example** (12, 18): GCF = 6 → LCM = (12 × 18) / 6 = **36** Multiples: 12, 24, 36… & 18, 36… → **36** is first common
✅ Pro Tip**: For two numbers: **LCM × GCF = a × b** — use to verify results.
⚠️ Avoid these common errors:
- LCM = product** — Only true if coprime (e.g., 5 and 7: LCM = 35). But 12 & 18 → 216 ≠ 36!
- Ignoring zero** — LCM(a, 0) is undefined (or 0 by some conventions); avoid.
- Using min instead of max exponents** — GCF uses min; LCM uses max.
- Negative numbers** — LCM uses absolute values: LCM(−12, 18) = LCM(12, 18) = 36
✅ Real-World Uses**:
- Scheduling**: Bus A (12 min), Bus B (15 min) → sync every **60 min**
- Fractions**: 1/4 + 1/6 → LCD = LCM(4,6) = **12**
- Gears**: 18-tooth & 24-tooth gears realign every **72** rotations
| Numbers | LCM | GCF | Use Case |
|---|---|---|---|
| 4, 6 | 12 | 2 | Fraction LCD |
| 7, 13 | 91 | 1 | Coprime (LCM = product) |
| 8, 12, 15 | 120 | 1 | 3-cycle sync |
| 16, 24 | 48 | 8 | Gear alignment |
📉 Efficiency Note**:
- LCM(1, n) = n
- If a | b, then LCM(a, b) = b (e.g., LCM(6, 18) = 18)
- LCM(a, b) ≥ max(a, b), equality iff one divides the other
➡️ Two Numbers
“LCM of 12 and 18?” → **36** (GCF = 6, 12×18/6 = 36)
➡️ Three+ Numbers
“LCM of 6, 8, 15?” → LCM(LCM(6,8),15) = LCM(24,15) = **120**
➡️ Prime Factorization
See Venn diagram: common primes (min = GCF), union (max = LCM)
Note: Handles up to 6 numbers. All inputs > 0. Output always ≥ 1.