If p,q,r,s,t be distinct primes and `N=pq^(2)r^(3)st`, then

Options

A. N has 96 divisors
B. N can be written as a product of two positive integers in 96 ways
C. N can be written as a product of two positive integers in 48 ways
D. N can not be divisible by 13!

Answer

To solve the problem, we need to determine the total number of divisors of the number \( N = p q^2 r^3 s t \), where \( p, q, r, s, t \) are distinct prime numbers.

### Step-by-Step Solution:

1. **Identify the Prime Factorization**:
The number \( N \) can be expressed in terms of its prime factors:
\[ N = p^1 \cdot q^2 \cdot r^3 \cdot s^1 \cdot t^1 \]
Here, the exponents of the prime factors are \( 1, 2, 3, 1, 1 \) respectively.

2. **Use the Divisor Formula**:
The formula for finding the total number of divisors \( d(N) \) of a number based on its prime factorization \( p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k} \) is given by:
\[ d(N) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \cdots (e_k + 1) \]
In our case, we have:
- For \( p \): \( e_1 = 1 \) → \( e_1 + 1 = 2 \)
- For \( q \): \( e_2 = 2 \) → \( e_2 + 1 = 3 \)
- For \( r \): \( e_3 = 3 \) → \( e_3 + 1 = 4 \)
- For \( s \): \( e_4 = 1 \) → \( e_4 + 1 = 2 \)
- For \( t \): \( e_5 = 1 \) → \( e_5 + 1 = 2 \)

3. **Calculate the Total Number of Divisors**:
Now, substituting these values into the divisor formula:
\[ d(N) = (1 + 1)(2 + 1)(3 + 1)(1 + 1)(1 + 1) = 2 \cdot 3 \cdot 4 \cdot 2 \cdot 2 \]

4. **Perform the Multiplication**:
Let's calculate it step by step:
- First, calculate \( 2 \cdot 3 = 6 \)
- Then, \( 6 \cdot 4 = 24 \)
- Next, \( 24 \cdot 2 = 48 \)
- Finally, \( 48 \cdot 2 = 96 \)

Therefore, the total number of divisors \( d(N) = 96 \).

5. **Conclusion**:
The total number of divisors of \( N = pq^2r^3st \) is **96**.