Product of divisors of 360 is :

Options

A. `(360)^12`
B. `(360)^120`
C. `(360)^22`
D. `6^24xx10^10`

Explanation

To find the product of the divisors of the number 360, we can follow these steps:

### Step 1: Prime Factorization of 360
First, we need to perform the prime factorization of 360.

- Divide 360 by 2:
- \( 360 \div 2 = 180 \)
- Divide 180 by 2:
- \( 180 \div 2 = 90 \)
- Divide 90 by 2:
- \( 90 \div 2 = 45 \)
- Divide 45 by 3:
- \( 45 \div 3 = 15 \)
- Divide 15 by 3:
- \( 15 \div 3 = 5 \)
- Finally, 5 is a prime number.

Thus, the prime factorization of 360 is:
\[ 360 = 2^3 \times 3^2 \times 5^1 \]

### Step 2: Finding the Number of Divisors
To find the number of divisors (factors) of a number based on its prime factorization, we use the formula:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \]
where \( e_1, e_2, e_3 \) are the powers of the prime factors.

For 360:
- The power of 2 is 3, so \( 3 + 1 = 4 \)
- The power of 3 is 2, so \( 2 + 1 = 3 \)
- The power of 5 is 1, so \( 1 + 1 = 2 \)

Now, we calculate:
\[ \text{Number of divisors} = 4 \times 3 \times 2 = 24 \]

### Step 3: Calculating the Product of Divisors
The product of the divisors of a number \( n \) can be calculated using the formula:
\[ \text{Product of divisors} = n^{\frac{t(n)}{2}} \]
where \( t(n) \) is the number of divisors.

From Step 2, we found that \( t(360) = 24 \).

Now, we calculate:
\[ \text{Product of divisors} = 360^{\frac{24}{2}} = 360^{12} \]

### Final Answer
Thus, the product of the divisors of 360 is:
\[ 360^{12} \]

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