Find the prime factorisation of the numbers :
`6000`

Options

A. `2 times 2times 2 times 2 times 3 times 5 times 5 `
B. `2 times 2times 2 times 2 times 2 times 5 times 5 `
C. `2 times 2times 2 times 2 times 3 times 5 times 7 `
D. `2 times 2times 2 times 2 times 3 times 3 times 5 `

Answer

To find the prime factorization of the number 6000, we will follow a systematic approach of dividing the number by its smallest prime factors until we reach 1. Here’s how we can do it step by step:

### Step 1: Start with the number 6000
We begin with the number 6000.

### Step 2: Divide by the smallest prime number
Since 6000 is an even number, we can divide it by 2 (the smallest prime number).
\[ 6000 \div 2 = 3000 \]

### Step 3: Continue dividing by 2
Now, we divide 3000 by 2 again since it is still even.
\[ 3000 \div 2 = 1500 \]

### Step 4: Divide by 2 again
Next, we divide 1500 by 2.
\[ 1500 \div 2 = 750 \]

### Step 5: Divide by 2 once more
We continue dividing by 2.
\[ 750 \div 2 = 375 \]

### Step 6: Divide by the next smallest prime number
375 is not even, so we try the next smallest prime number, which is 3. We check if 375 is divisible by 3:
\[ 3 + 7 + 5 = 15 \quad (\text{which is divisible by } 3) \]
Now, we divide:
\[ 375 \div 3 = 125 \]

### Step 7: Divide by 5
Next, we divide 125 by 5 (the next smallest prime number).
\[ 125 \div 5 = 25 \]

### Step 8: Divide by 5 again
We divide 25 by 5.
\[ 25 \div 5 = 5 \]

### Step 9: Divide by 5 one last time
Finally, we divide 5 by 5.
\[ 5 \div 5 = 1 \]

### Step 10: Write down the prime factors
Now, we can write down all the prime factors we used:
- We divided by 2 four times: \( 2 \times 2 \times 2 \times 2 \) or \( 2^4 \)
- We divided by 3 once: \( 3 \)
- We divided by 5 three times: \( 5 \times 5 \times 5 \) or \( 5^3 \)

### Final Prime Factorization
Putting it all together, the prime factorization of 6000 is:
\[ 6000 = 2^4 \times 3^1 \times 5^3 \]