The number of ways to choose 7 distinct natural numbers from the first 100 natural numbers such that any two chosen numbers differ atleast by 7 can be expressed as `""^(n)C_(7)`. Find the number of divisors of n.

Answer

To solve the problem of choosing 7 distinct natural numbers from the first 100 natural numbers such that any two chosen numbers differ by at least 7, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to select 7 distinct natural numbers from the set {1, 2, ..., 100} such that the difference between any two selected numbers is at least 7.

2. **Transforming the Selection**:
To ensure that any two chosen numbers differ by at least 7, we can redefine our selection. If we select a number \( x_i \), then the next number \( x_{i+1} \) must be at least \( x_i + 7 \).

3. **Adjusting the Range**:
To simplify the selection, we can make a change of variables. Let:
- \( y_1 = x_1 \)
- \( y_2 = x_2 - 7 \)
- \( y_3 = x_3 - 14 \)
- \( y_4 = x_4 - 21 \)
- \( y_5 = x_5 - 28 \)
- \( y_6 = x_6 - 35 \)
- \( y_7 = x_7 - 42 \)

This transformation ensures that \( y_1, y_2, ..., y_7 \) are distinct natural numbers chosen from the adjusted range.

4. **Determining the New Range**:
The maximum value for \( x_7 \) is 100, which means:
\[ y_7 = x_7 - 42 \leq 100 - 42 = 58 \]
Therefore, \( y_1, y_2, ..., y_7 \) must be chosen from the set {1, 2, ..., 58}.

5. **Counting the Selections**:
Now, we need to select 7 distinct numbers from the 58 available numbers. The number of ways to do this is given by the binomial coefficient:
\[ \binom{58}{7} \]

6. **Expressing in Terms of n**:
The problem states that this can be expressed as \( \binom{n}{7} \). Thus, we have:
\[ n = 58 \]

7. **Finding the Number of Divisors of n**:
To find the number of divisors of \( n = 58 \), we first find its prime factorization:
\[ 58 = 2 \times 29 \]
The prime factorization shows that \( 58 \) has the form \( p_1^{e_1} \times p_2^{e_2} \), where \( p_1 = 2 \) and \( p_2 = 29 \), both raised to the power of 1.

The formula for the number of divisors is:
\[ (e_1 + 1)(e_2 + 1) = (1 + 1)(1 + 1) = 2 \times 2 = 4 \]

### Final Answer:
The number of divisors of \( n \) is **4**.