Find the LCM of 128 and 208 by prime factorisation method

Answer

To find the LCM (Least Common Multiple) of 128 and 208 using the prime factorization method, we will follow these steps:

### Step 1: Find the Prime Factorization of Both Numbers

**For 128:**
- Divide by 2: \(128 \div 2 = 64\)
- Divide by 2: \(64 \div 2 = 32\)
- Divide by 2: \(32 \div 2 = 16\)
- Divide by 2: \(16 \div 2 = 8\)
- Divide by 2: \(8 \div 2 = 4\)
- Divide by 2: \(4 \div 2 = 2\)
- Divide by 2: \(2 \div 2 = 1\)

So, the prime factorization of 128 is:
\[ 128 = 2^7 \]

**For 208:**
- Divide by 2: \(208 \div 2 = 104\)
- Divide by 2: \(104 \div 2 = 52\)
- Divide by 2: \(52 \div 2 = 26\)
- Divide by 2: \(26 \div 2 = 13\)
- Divide by 13: \(13 \div 13 = 1\)

So, the prime factorization of 208 is:
\[ 208 = 2^4 \times 13^1 \]

### Step 2: Identify the Highest Power of Each Prime Factor

Now we will identify the highest power of each prime factor from both factorizations:
- For the prime factor 2: The highest power is \(2^7\) (from 128).
- For the prime factor 13: The highest power is \(13^1\) (from 208).

### Step 3: Multiply the Highest Powers of All Prime Factors

To find the LCM, we multiply the highest powers of all prime factors:
\[ \text{LCM} = 2^7 \times 13^1 \]

Calculating this:
\[ \text{LCM} = 128 \times 13 \]
\[ \text{LCM} = 1664 \]

Thus, the LCM of 128 and 208 is **1664**.

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