To find the least common multiple (LCM) of two numbers, you can use the prime factorization method or the formula: \[ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)} \] where GCD is the greatest common divisor. ### Method 1: Prime Factorization 1. **Prime factorization:** - 12 can be factored into \(2^2 \times 3^1\) - 9 can be factored into \(3^2\) 2. **Take the highest power of each prime:** - For 2: \(2^2\) - For 3: \(3^2\) 3. **Multiply these together:** \[ LCM = 2^2 \times 3^2 = 4 \times 9 = 36 \] ### Method 2: Using GCD 1. **Calculate GCD of 12 and 9:** - The factors of 12 are \(1, 2, 3, 4, 6, 12\) - The factors of 9 are \(1, 3, 9\) - The GCD is \(3\). 2. **Use the LCM formula:** \[ LCM(12, 9) = \frac{|12 \times 9|}{GCD(12, 9)} = \frac{108}{3} = 36 \] In both methods, the least common multiple of 12 and 9 is **36**.

the lcm of 12 of 9

1 answer

Bot GPT 4-o mini answered
2 hours ago

To find the least common multiple (LCM) of two numbers, you can use the prime factorization method or the formula:

\[ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)} \]

where GCD is the greatest common divisor.

Method 1: Prime Factorization

Prime factorization:
- 12 can be factored into \(2^2 \times 3^1\)
- 9 can be factored into \(3^2\)
Take the highest power of each prime:
- For 2: \(2^2\)
- For 3: \(3^2\)
Multiply these together: \[ LCM = 2^2 \times 3^2 = 4 \times 9 = 36 \]

Method 2: Using GCD

Calculate GCD of 12 and 9:
- The factors of 12 are \(1, 2, 3, 4, 6, 12\)
- The factors of 9 are \(1, 3, 9\)
- The GCD is \(3\).
Use the LCM formula: \[ LCM(12, 9) = \frac{|12 \times 9|}{GCD(12, 9)} = \frac{108}{3} = 36 \]

In both methods, the least common multiple of 12 and 9 is 36.