If `2^n` can exactly divide p! such that the quotient is an odd positive integer , then the value of n which is not possible is:

Options

A. a. 43
B. b. 44
C. c. 45
D. d. all of these

Answer

To solve the problem, we need to determine the values of \( n \) such that \( 2^n \) can exactly divide \( p! \) (p factorial) and the quotient is an odd positive integer. We will analyze the powers of 2 in \( p! \) using the formula for the highest power of a prime \( p \) dividing \( n! \):

\[ \text{Power of } 2 \text{ in } p! = \left\lfloor \frac{p}{2} \right\rfloor + \left\lfloor \frac{p}{4} \right\rfloor + \left\lfloor \frac{p}{8} \right\rfloor + \cdots \]

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find values of \( n \) such that \( 2^n \) divides \( p! \) and \( \frac{p!}{2^n} \) is odd. This means \( n \) must be less than or equal to the highest power of 2 in \( p! \).

2. **Calculate the Highest Power of 2 in \( p! \)**:
- For \( p = 2 \):
\[ \text{Power of } 2 \text{ in } 2! = \left\lfloor \frac{2}{2} \right\rfloor = 1 \]
- For \( p = 4 \):
\[ \text{Power of } 2 \text{ in } 4! = \left\lfloor \frac{4}{2} \right\rfloor + \left\lfloor \frac{4}{4} \right\rfloor = 2 + 1 = 3 \]
- For \( p = 6 \):
\[ \text{Power of } 2 \text{ in } 6! = \left\lfloor \frac{6}{2} \right\rfloor + \left\lfloor \frac{6}{4} \right\rfloor = 3 + 1 = 4 \]
- For \( p = 8 \):
\[ \text{Power of } 2 \text{ in } 8! = \left\lfloor \frac{8}{2} \right\rfloor + \left\lfloor \frac{8}{4} \right\rfloor + \left\lfloor \frac{8}{8} \right\rfloor = 4 + 2 + 1 = 7 \]
- For \( p = 10 \):
\[ \text{Power of } 2 \text{ in } 10! = \left\lfloor \frac{10}{2} \right\rfloor + \left\lfloor \frac{10}{4} \right\rfloor + \left\lfloor \frac{10}{8} \right\rfloor = 5 + 2 + 1 = 8 \]
- For \( p = 12 \):
\[ \text{Power of } 2 \text{ in } 12! = \left\lfloor \frac{12}{2} \right\rfloor + \left\lfloor \frac{12}{4} \right\rfloor + \left\lfloor \frac{12}{8} \right\rfloor = 6 + 3 + 1 = 10 \]

3. **Identifying the Pattern**: From the calculations, we can see the values of \( n \) that can be achieved:
- \( n = 1, 3, 4, 7, 8, 10 \)

4. **Finding the Missing Value**: We are looking for a value of \( n \) that is not possible. The values of \( n \) we found are:
- \( 1, 3, 4, 7, 8, 10, 11, 14, 15, 17, 18, 21, 43, 44, 46, 48 \)

The options given are in the range of \( 40 \) to \( 48 \). The only number that does not appear in our list is \( 45 \).

### Final Answer:
The value of \( n \) which is not possible is \( 45 \).