Find the prime factorisation of the numbers given below:
127008

Answer

To find the prime factorization of the number 127008, we will systematically divide the number by the smallest prime numbers until we can no longer divide it. Here’s how we can do it step by step:

### Step 1: Start with the smallest prime number, which is 2.
- Since 127008 is even, we can divide it by 2.
- \( 127008 ÷ 2 = 63504 \)

### Step 2: Continue dividing by 2.
- \( 63504 ÷ 2 = 31752 \)
- \( 31752 ÷ 2 = 15876 \)
- \( 15876 ÷ 2 = 7938 \)
- \( 7938 ÷ 2 = 3969 \)

### Step 3: Check if we can divide by 2 again.
- 3969 is odd, so we cannot divide by 2 anymore. Now we will try the next prime number, which is 3.

### Step 4: Divide by 3.
- \( 3969 ÷ 3 = 1323 \)
- \( 1323 ÷ 3 = 441 \)
- \( 441 ÷ 3 = 147 \)
- \( 147 ÷ 3 = 49 \)

### Step 5: Now, check for the next prime number.
- 49 is not divisible by 3, so we will try the next prime number, which is 7.

### Step 6: Divide by 7.
- \( 49 ÷ 7 = 7 \)
- \( 7 ÷ 7 = 1 \)

### Step 7: Compile the prime factors.
- We have divided 127008 as follows:
- \( 2^5 \) (since we divided by 2 five times)
- \( 3^4 \) (since we divided by 3 four times)
- \( 7^2 \) (since we divided by 7 two times)

### Final Prime Factorization:
Thus, the prime factorization of 127008 is:
\[ 127008 = 2^5 \times 3^4 \times 7^2 \]

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