Find the LCM of 32 and 28 by prime factorisation

Answer

To find the LCM (Least Common Multiple) of 32 and 28 using prime factorization, we will follow these steps:

### Step 1: Prime Factorization of 32
- Start by dividing 32 by the smallest prime number, which is 2.
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1

So, the prime factorization of 32 is:
\[ 32 = 2^5 \]

### Step 2: Prime Factorization of 28
- Now, let's factor 28 using the smallest prime number, which is again 2.
- 28 ÷ 2 = 14
- 14 ÷ 2 = 7
- 7 ÷ 7 = 1

So, the prime factorization of 28 is:
\[ 28 = 2^2 \times 7^1 \]

### Step 3: Identify the Highest Powers of Each Prime Factor
- From the prime factorizations, we have:
- For the prime number 2: the highest power is \( 2^5 \) (from 32).
- For the prime number 7: the highest power is \( 7^1 \) (from 28).

### Step 4: Calculate the LCM
- The LCM is found by multiplying the highest powers of all prime factors together:
\[ \text{LCM} = 2^5 \times 7^1 \]

Calculating this:
- \( 2^5 = 32 \)
- \( 7^1 = 7 \)

Now, multiply these together:
\[ \text{LCM} = 32 \times 7 = 224 \]

### Final Answer
The LCM of 32 and 28 is **224**.

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