Find the prime factorisation of the numbers given below:
340200

Answer

To find the prime factorization of the number 340200, we will follow these steps:

### Step 1: Divide by the smallest prime number
Start with the number 340200. Since it is even, we can divide it by 2.

340200 ÷ 2 = 170100

### Step 2: Continue dividing by 2
Next, we can divide 170100 by 2 again since it is still even.

170100 ÷ 2 = 85050

### Step 3: Divide by 2 once more
Now, we divide 85050 by 2 again.

85050 ÷ 2 = 42525

### Step 4: Move to the next smallest prime number
42525 is not even, so we try dividing by the next smallest prime number, which is 3.

42525 ÷ 3 = 14175

### Step 5: Continue dividing by 3
Now we divide 14175 by 3 again.

14175 ÷ 3 = 4725

### Step 6: Divide by 3 again
Next, we divide 4725 by 3.

4725 ÷ 3 = 1575

### Step 7: Divide by 3 once more
Now, we divide 1575 by 3.

1575 ÷ 3 = 525

### Step 8: Divide by 3 again
Next, we divide 525 by 3.

525 ÷ 3 = 175

### Step 9: Move to the next prime number
Now, we divide 175 by the next smallest prime number, which is 5.

175 ÷ 5 = 35

### Step 10: Divide by 5 again
Next, we divide 35 by 5.

35 ÷ 5 = 7

### Step 11: Final prime number
Now we have reached 7, which is a prime number.

### Step 12: Write the prime factorization
Now we can write the prime factorization of 340200. We have:

- 2 appears 3 times
- 3 appears 5 times
- 5 appears 2 times
- 7 appears 1 time

Thus, the prime factorization of 340200 is:

\[ 340200 = 2^3 \times 3^5 \times 5^2 \times 7^1 \]