Product of divisors of 7056 is :

Options

A. `(84)^48`
B. `(84)^44`
C. `(84)^45`
D. none of these

Explanation

To find the product of the divisors of the number 7056, we can follow these steps:

### Step 1: Prime Factorization of 7056
We start by factorizing 7056 into its prime factors.

1. Divide by 2:
- 7056 ÷ 2 = 3528
2. Divide by 2 again:
- 3528 ÷ 2 = 1764
3. Divide by 2 again:
- 1764 ÷ 2 = 882
4. Divide by 2 again:
- 882 ÷ 2 = 441
5. Now, divide by 3:
- 441 ÷ 3 = 147
6. Divide by 3 again:
- 147 ÷ 3 = 49
7. Finally, divide by 7:
- 49 ÷ 7 = 7
8. Divide by 7 again:
- 7 ÷ 7 = 1

So, the prime factorization of 7056 is:
\[ 7056 = 2^4 \times 3^2 \times 7^2 \]

### Step 2: Finding the Number of Divisors
To find the number of divisors \( d(N) \) of a number \( N \) with the prime factorization \( p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k} \), we use the formula:
\[ d(N) = (e_1 + 1)(e_2 + 1) \ldots (e_k + 1) \]

For 7056:
- The powers of the prime factors are 4 (for 2), 2 (for 3), and 2 (for 7).
- Therefore:
\[ d(7056) = (4 + 1)(2 + 1)(2 + 1) = 5 \times 3 \times 3 = 45 \]

### Step 3: Product of the Divisors
The product of the divisors of a number \( N \) can be calculated using the formula:
\[ \text{Product of divisors} = N^{d(N)/2} \]

For 7056:
- We already found \( d(7056) = 45 \).
- Therefore:
\[ \text{Product of divisors} = 7056^{45/2} = 7056^{22.5} \]

### Step 4: Finding the Square Root of 7056
Next, we need to find the square root of 7056:
\[ \sqrt{7056} = 84 \]

### Step 5: Final Calculation
Now, we can express the product of the divisors:
\[ \text{Product of divisors} = 84^{45} \]

### Conclusion
Thus, the product of the divisors of 7056 is:
\[ 84^{45} \]

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