The question given below is followed by two statements, A and B. Mark the answer using the following instructions:
Q. N is a natural number that has exactly 24 factors. What is the number of factors of N3?
A. When N is multipled by 3, the resultant number has 32 factors.
B. When N is multipled by 5, the resultant number has 30 factors.

Options

A. Mark (a) if the question can be answered by using one of the statements alone, but cannot be answered by using the other statement alone.
B. Mark (b) if the question can be answered by using either statements alone.
C. Mark (c) if the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
D. Mark (d) if the question cannot be answered even by using both the statements together.

Answer

To solve the problem, we need to determine the number of factors of \( N^3 \) given that \( N \) has exactly 24 factors. We will analyze the statements A and B to see how they help us.

### Step-by-Step Solution:

1. **Understanding the Number of Factors**:
The number of factors of a number \( N \) can be determined from its prime factorization. If \( N \) can be expressed as:
\[ N = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k} \]
then the number of factors \( d(N) \) is given by:
\[ d(N) = (e_1 + 1)(e_2 + 1) \ldots (e_k + 1) \]
In this case, we know \( d(N) = 24 \).

2. **Finding Possible Prime Factorizations**:
Since \( d(N) = 24 \), we can express 24 in terms of its factors:
- \( 24 = 24 \) (one prime factor)
- \( 24 = 12 \times 2 \) (two prime factors)
- \( 24 = 8 \times 3 \) (two prime factors)
- \( 24 = 6 \times 4 \) (two prime factors)
- \( 24 = 4 \times 3 \times 2 \) (three prime factors)

For example, if \( N = p_1^5 \times p_2^3 \), then \( d(N) = (5 + 1)(3 + 1) = 6 \times 4 = 24 \).

3. **Calculating \( d(N^3) \)**:
If \( N \) has the prime factorization \( p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k} \), then:
\[ N^3 = p_1^{3e_1} \times p_2^{3e_2} \times \ldots \times p_k^{3e_k} \]
The number of factors of \( N^3 \) is:
\[ d(N^3) = (3e_1 + 1)(3e_2 + 1) \ldots (3e_k + 1) \]

4. **Using Statement A**:
Statement A tells us that when \( N \) is multiplied by 3, the resultant number has 32 factors. This means:
\[ d(N \times 3) = d(N) \times d(3) = d(N) \times 2 \]
Since \( d(N) = 24 \):
\[ d(N \times 3) = 24 \times 2 = 48 \]
However, we need to analyze how this affects \( d(N^3) \).

5. **Using Statement B**:
Statement B states that when \( N \) is multiplied by 5, the resultant number has 30 factors. This means:
\[ d(N \times 5) = d(N) \times d(5) = d(N) \times 2 \]
Again, since \( d(N) = 24 \):
\[ d(N \times 5) = 24 \times 2 = 48 \]
This does not directly help us find \( d(N^3) \).

6. **Conclusion**:
From the analysis, we can conclude that we can find the number of factors of \( N^3 \) using Statement A alone. Therefore, the answer is:
- **Option A**: The question can be answered by using one of the statements alone, but cannot be answered by using the other statement.