Find the LCM of 8,12 and 15.

Answer

To find the LCM (Least Common Multiple) of the numbers 8, 12, and 15, we can use the method of prime factorization. Here’s a step-by-step solution:

### Step 1: Prime Factorization of Each Number
- **8**: The prime factorization of 8 is \(2^3\) (since \(8 = 2 \times 2 \times 2\)).
- **12**: The prime factorization of 12 is \(2^2 \times 3^1\) (since \(12 = 2 \times 2 \times 3\)).
- **15**: The prime factorization of 15 is \(3^1 \times 5^1\) (since \(15 = 3 \times 5\)).

### Step 2: Identify the Highest Powers of Each Prime Factor
Next, we will identify the highest power of each prime factor that appears in the factorizations:
- For the prime factor **2**, the highest power is \(2^3\) (from 8).
- For the prime factor **3**, the highest power is \(3^1\) (from both 12 and 15).
- For the prime factor **5**, the highest power is \(5^1\) (from 15).

### Step 3: Multiply the Highest Powers Together
Now, we will multiply these highest powers together to find the LCM:
\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 \]
Calculating this step-by-step:
- \(2^3 = 8\)
- \(3^1 = 3\)
- \(5^1 = 5\)

Now, multiply these values:
\[ 8 \times 3 = 24 \]
\[ 24 \times 5 = 120 \]

### Final Answer
Thus, the LCM of 8, 12, and 15 is **120**.

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