Find the LCM of 8, 12 and 16 using prime factorisation

Answer

To find the LCM (Least Common Multiple) of the numbers 8, 12, and 16 using prime factorization, we can follow these steps:

### Step 1: Prime Factorization of Each Number
- **For 8**:
- 8 can be expressed as \(2 \times 2 \times 2\) or \(2^3\).

- **For 12**:
- 12 can be expressed as \(2 \times 2 \times 3\) or \(2^2 \times 3^1\).

- **For 16**:
- 16 can be expressed as \(2 \times 2 \times 2 \times 2\) or \(2^4\).

### Step 2: Identify the Highest Power of Each Prime Factor
From the prime factorizations:
- The prime factors we have are 2 and 3.
- The highest power of 2 from the factorizations is \(2^4\) (from 16).
- The highest power of 3 is \(3^1\) (from 12).

### Step 3: Calculate the LCM
To find the LCM, we multiply the highest powers of all prime factors together:
\[ \text{LCM} = 2^4 \times 3^1 \]
Calculating this:
\[ 2^4 = 16 \quad \text{and} \quad 3^1 = 3 \]
Now multiply these:
\[ \text{LCM} = 16 \times 3 = 48 \]

### Final Answer
The LCM of 8, 12, and 16 is **48**.

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