Find the prime factorisation of the numbers given below:
405000

Answer

To find the prime factorization of the number 405000, we will follow these steps:

### Step 1: Divide by 2
Since the last digit of 405000 is even, we can divide it by 2.
- \( 405000 \div 2 = 202500 \)

### Step 2: Divide by 2 again
The last digit of 202500 is also even, so we divide it by 2 again.
- \( 202500 \div 2 = 101250 \)

### Step 3: Divide by 2 again
The last digit of 101250 is still even, so we divide it by 2 once more.
- \( 101250 \div 2 = 50625 \)

### Step 4: Check divisibility by 3
Now, we check 50625. The last digit is not even, so we add the digits to check for divisibility by 3:
- Sum of digits: \( 5 + 0 + 6 + 2 + 5 = 18 \) (which is divisible by 3)
- Now, divide by 3:
- \( 50625 \div 3 = 16875 \)

### Step 5: Divide by 3 again
Next, we check 16875 for divisibility by 3:
- Sum of digits: \( 1 + 6 + 8 + 7 + 5 = 27 \) (which is divisible by 3)
- Divide by 3:
- \( 16875 \div 3 = 5625 \)

### Step 6: Divide by 3 again
Now check 5625:
- Sum of digits: \( 5 + 6 + 2 + 5 = 18 \) (which is divisible by 3)
- Divide by 3:
- \( 5625 \div 3 = 1875 \)

### Step 7: Divide by 3 again
Next, check 1875:
- Sum of digits: \( 1 + 8 + 7 + 5 = 21 \) (which is divisible by 3)
- Divide by 3:
- \( 1875 \div 3 = 625 \)

### Step 8: Check divisibility by 5
Now, we check 625. The last digit is 5, so we can divide by 5:
- \( 625 \div 5 = 125 \)

### Step 9: Divide by 5 again
Next, divide 125 by 5:
- \( 125 \div 5 = 25 \)

### Step 10: Divide by 5 again
Now, divide 25 by 5:
- \( 25 \div 5 = 5 \)

### Step 11: Divide by 5 one last time
Finally, divide 5 by 5:
- \( 5 \div 5 = 1 \)

### Summary of Factors
Now we can summarize the prime factors:
- We divided by 2 three times: \( 2^3 \)
- We divided by 3 four times: \( 3^4 \)
- We divided by 5 four times: \( 5^4 \)

### Final Prime Factorization
Thus, the prime factorization of 405000 is:
\[ 2^3 \times 3^4 \times 5^4 \]

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