Question

the lcm of 12 of 9

Answer 1

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**.