To solve the problem, we need to find the probability that the first marker drawn is even and the second marker drawn is odd. Then, we will determine the number of divisors of the resulting value \( k \).
### Step-by-Step Solution:
1. **Identify the total number of markers**:
The bag contains markers numbered from 1 to 17. Therefore, the total number of markers is 17.
2. **Count the even markers**:
The even numbers between 1 and 17 are: 2, 4, 6, 8, 10, 12, 14, 16.
Thus, there are 8 even markers.
3. **Count the odd markers**:
The odd numbers between 1 and 17 are: 1, 3, 5, 7, 9, 11, 13, 15, 17.
Thus, there are 9 odd markers.
4. **Calculate the probability of drawing an even marker first**:
The probability of drawing an even marker first is given by:
\[
P(\text{Even first}) = \frac{\text{Number of even markers}}{\text{Total markers}} = \frac{8}{17}
\]
5. **Calculate the probability of drawing an odd marker second**:
Since the first marker is replaced, the total number of markers remains the same. The probability of drawing an odd marker second is:
\[
P(\text{Odd second}) = \frac{\text{Number of odd markers}}{\text{Total markers}} = \frac{9}{17}
\]
6. **Calculate the combined probability**:
The combined probability of the first marker being even and the second marker being odd is:
\[
P(\text{Even first and Odd second}) = P(\text{Even first}) \times P(\text{Odd second}) = \frac{8}{17} \times \frac{9}{17} = \frac{72}{289}
\]
7. **Identify the value of \( k \)**:
From the probability calculated, we have:
\[
\frac{72}{289} = \frac{k}{289}
\]
Thus, \( k = 72 \).
8. **Find the number of divisors of \( k \)**:
To find the number of divisors of \( k = 72 \), we first find its prime factorization:
\[
72 = 2^3 \times 3^2
\]
The formula for finding the number of divisors from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \) is:
\[
(e_1 + 1)(e_2 + 1)
\]
For \( 72 = 2^3 \times 3^2 \):
- \( e_1 = 3 \) (for prime 2)
- \( e_2 = 2 \) (for prime 3)
Therefore, the number of divisors is:
\[
(3 + 1)(2 + 1) = 4 \times 3 = 12
\]
### Final Answer:
The number of divisors of \( k \) is **12**.
