LCM of 9 and 12 = 36
Quick Answer
LCM(9 and 12) = 36.
First common multiples: 36, 72, 108, 144, 180. Related GCF: 3.
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Step-by-Step: LCM via the GCF Identity
The fastest way to compute the LCM uses the identity LCM(a, b) × GCF(a, b) = a × b. So LCM = (a × b) / GCF. We first find the GCF with the Euclidean algorithm, then apply the identity.
Euclidean reduction for (9, 12):
| Step | Dividend ÷ Divisor | Quotient | Remainder |
|---|---|---|---|
| 1 | 12 ÷ 9 | 1 | 3 |
| 2 | 9 ÷ 3 | 3 | 0 |
GCF = 3. Applying the identity:
LCM = (9 × 12) ÷ GCF = 108 ÷ 3 = 36
LCM(9 and 12) = 36
First 5 Common Multiples of 9 and 12
Every common multiple is a multiple of the LCM. The first five are:
| k | k × LCM | Value |
|---|---|---|
| 1 | 1 × 36 | 36 |
| 2 | 2 × 36 | 72 |
| 3 | 3 × 36 | 108 |
| 4 | 4 × 36 | 144 |
| 5 | 5 × 36 | 180 |
Related: GCF of 9 and 12
GCF(9, 12) = 3
From the identity LCM × GCF = a × b:
36 × 3 = 108 = 9 × 12
See the dedicated GCF of 9 and 12 page for the full common-divisor list and Euclidean walk-through.
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How to Find the Least Common Multiple
The Least Common Multiple (LCM) of two or more positive integers is the smallest positive integer divisible by each of them.
Method 1: GCF Identity (used above)
- Find the GCF (Greatest Common Factor) using the Euclidean algorithm.
- Apply the identity: LCM = (a × b) / GCF.
Method 2: Prime Factorization
- Find the prime factorization of each number.
- For each prime that appears in any factorization, take the highest exponent across all inputs.
- Multiply these prime powers together.
Example: LCM(12, 18) via prime factorization
- 12 = 2² × 3¹
- 18 = 2¹ × 3²
- Take the highest exponent of each prime: 2² (from 12), 3² (from 18).
- LCM = 2² × 3² = 4 × 9 = 36
Method 3: Listing Multiples
- List the multiples of each number until you find a common one.
- That smallest common multiple is the LCM.
This is the slowest method for large inputs; the GCF identity is preferred.
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Frequently Asked Questions
What is the LCM of 9 and 12?
LCM(9 and 12) = 36. This is the smallest positive integer divisible by each input.
How is the LCM of 9 and 12 calculated?
Apply the identity LCM = (a × b) / GCF. For 9 and 12, GCF = 3, so LCM = 108 / 3 = 36.
How are LCM and GCF related?
By the identity LCM(a, b) × GCF(a, b) = a × b. So once you have the GCF (via the Euclidean algorithm), the LCM follows immediately: LCM = (a × b) / GCF.
What is LCM used for?
Two major uses: (1) adding/subtracting fractions with different denominators (find the common denominator); (2) scheduling — when two repeating events occur together (every 12 days and every 18 days → coincide every LCM(12, 18) = 36 days).
Related Calculators
- GCF of 9 and 12 = 3
- Prime Factorization Calculator
- Divisors of a Number
- Multiples of a Number
- Fraction Calculator (uses LCM for common denominators)
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