LCM of 9 and 12 with Working
The LCM of 9 and 12 (least common multiple) is 36. 36 = 2² × 3², and the greatest common factor (HCF) of 9 and 12 is 3.
36 ÷ 9 = 4 · 36 ÷ 12 = 3 · GCF = 3
Open 9 and 12 in the calculator
How to find the LCM of 9 and 12: six methods
Every method below reaches 36. Pick the one your class uses; the working is the same as the calculator shows.
Listing multiples: LCM of 9 and 12
Listing multiples: LCM(9, 12)
Write out the multiples of each number until one multiple appears in every row. The first number that appears in all rows is the least common multiple.
36 is the first multiple shared by every number.
LCM(9, 12) = 36
Prime factorization: LCM of 9 and 12
Prime factorization: LCM(9, 12)
Break each number into prime factors.
- 9 = 3 × 3 = 3²
- 12 = 2 × 2 × 3 = 2² × 3
Take the highest power of every prime that appears, then multiply them together.
| Prime | Highest power | From |
|---|---|---|
| 2 | 2² | 12 |
| 3 | 3² | 9 |
LCM = 2² × 3² = 36
LCM(9, 12) = 36
Cake (ladder) method: LCM of 9 and 12
Cake (ladder) method: LCM(9, 12)
Write the numbers in a row. Divide by a prime that goes into at least two of them; numbers it does not divide are carried down unchanged. Repeat until no prime divides two or more numbers.
| 3 | 9 | 12 |
|---|---|---|
| 3 | 4 |
Multiply the divisors on the left by the numbers left in the bottom row:
LCM = 3 × 3 × 4 = 36
LCM(9, 12) = 36
Division method: LCM of 9 and 12
Division method: LCM(9, 12)
Divide the row by the smallest prime that goes into at least one number; carry the rest down. Keep going until every number becomes 1. The LCM is the product of all the divisors on the left.
| 2 | 9 | 12 |
|---|---|---|
| 2 | 9 | 6 |
| 3 | 9 | 3 |
| 3 | 3 | 1 |
| 1 | 1 |
LCM = 2 × 2 × 3 × 3 = 2² × 3² = 36
LCM(9, 12) = 36
Using the GCF: LCM of 9 and 12
Using the GCF: LCM(9, 12)
For two numbers, LCM(a, b) = a × b ÷ GCF(a, b).
- GCF(9, 12) = 3, so LCM(9, 12) = 9 × 12 ÷ 3 = 108 ÷ 3 = 36
LCM(9, 12) = 36
Venn diagram: LCM of 9 and 12
Venn diagram: LCM(9, 12)
Put each number's prime factors in its circle. Factors shared by two (or three) numbers go where the circles overlap, written once. Multiply every factor in the diagram to get the LCM; the overlap alone gives the GCF.
LCM = 3 × 3 × 2 × 2 = 36
GCF = 3 = 3
LCM(9, 12) = 36
Facts about 9 and 12
- Prime factorization: 9 = 3²; 12 = 2² × 3.
- GCF (HCF) of 9 and 12: 3. LCM × GCF = 108 = 9 × 12.
- First three common multiples of 9 and 12: 36, 72, 108.
- 9 and 12 are not coprime: both share the factor 3.
Nearby pairs
- LCM of 8 and 12 = 24
- LCM of 6 and 9 = 18
- LCM of 7 and 9 = 63
- LCM of 8 and 9 = 72
Frequently asked questions
What is the LCM of 9 and 12?
The LCM of 9 and 12 is 36. 36 is the smallest positive number that both 9 and 12 divide into exactly: 36 ÷ 9 = 4 and 36 ÷ 12 = 3.
What is the GCF of 9 and 12?
The GCF (HCF) of 9 and 12 is 3. Check: LCM × GCF = 36 × 3 = 108 = 9 × 12.
How do you find the LCM of 9 and 12?
Prime factorization: 9 = 3² and 12 = 2² × 3. Take each prime at its highest power: 2² × 3² = 36.
Sources
- Wikipedia: Least common multiple: definition, notation, LCM × GCD = a × b
- Wolfram MathWorld: Least Common Multiple: formal properties
- Common Core 6.NS.B.4: GCF and LCM in grade 6