The number of division of `2^6. 3^5 . 5^3. 7^4` 11 is equal to

Options

A. `11^2-1`
B. `21^2-1`
C. `31^2-1`
D. `41^2-1`

Answer

To find the number of divisors of the number \(2^6 \cdot 3^5 \cdot 5^3 \cdot 7^4 \cdot 11^1\), we can use the formula for finding the number of divisors based on the prime factorization of a number.

### Step-by-Step Solution:

1. **Identify the prime factorization**:
The number is given as \(2^6 \cdot 3^5 \cdot 5^3 \cdot 7^4 \cdot 11^1\). Here, the prime factors are \(2\), \(3\), \(5\), \(7\), and \(11\) with their respective powers.

2. **Use the divisor formula**:
The formula for finding the number of divisors \(d(n)\) of a number \(n\) which is expressed in its prime factorization form \(p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k}\) is:
\[ d(n) = (e_1 + 1)(e_2 + 1) \cdots (e_k + 1) \]
where \(e_i\) are the powers of the prime factors.

3. **Apply the formula**:
For our number:
- For \(2^6\), \(e_1 = 6\) → \(6 + 1 = 7\)
- For \(3^5\), \(e_2 = 5\) → \(5 + 1 = 6\)
- For \(5^3\), \(e_3 = 3\) → \(3 + 1 = 4\)
- For \(7^4\), \(e_4 = 4\) → \(4 + 1 = 5\)
- For \(11^1\), \(e_5 = 1\) → \(1 + 1 = 2\)

4. **Multiply the results**:
Now, we multiply these results together:
\[ d(n) = 7 \cdot 6 \cdot 4 \cdot 5 \cdot 2 \]

5. **Calculate the product**:
- First, calculate \(7 \cdot 6 = 42\)
- Next, calculate \(42 \cdot 4 = 168\)
- Then, calculate \(168 \cdot 5 = 840\)
- Finally, calculate \(840 \cdot 2 = 1680\)

Thus, the total number of divisors of the number \(2^6 \cdot 3^5 \cdot 5^3 \cdot 7^4 \cdot 11^1\) is \(1680\).

### Final Answer:
The number of divisors is \(1680\).