A bag contains 17 markers with numbers 1 to 17. A marker is drawn at random and then replaced, a second marker is drawn then the probability that first number is even and second is odd is k/289 then number of divisors of k is______.

Answer

To solve the problem step by step, we need to calculate the probability that the first marker drawn is even and the second marker drawn is odd. Then we will find the value of \( k \) and determine the number of divisors of \( k \).

### Step 1: Identify the total number of markers
The bag contains markers numbered from 1 to 17. Therefore, the total number of markers is:
\[ n = 17 \]

### Step 2: Count the even and odd markers
Among the markers numbered 1 to 17:
- The even markers are: 2, 4, 6, 8, 10, 12, 14, 16 (Total = 8)
- The odd markers are: 1, 3, 5, 7, 9, 11, 13, 15, 17 (Total = 9)

### Step 3: Calculate the probability of drawing an even marker first
The probability of drawing an even marker first is given by:
\[ P(\text{Even first}) = \frac{\text{Number of even markers}}{\text{Total markers}} = \frac{8}{17} \]

### Step 4: Calculate the probability of drawing an odd marker second
Since the first marker is replaced, the total number of markers remains the same for the second draw. The probability of drawing an odd marker second is:
\[ P(\text{Odd second}) = \frac{\text{Number of odd markers}}{\text{Total markers}} = \frac{9}{17} \]

### Step 5: Calculate the combined probability
The combined probability of the first marker being even and the second marker being odd is:
\[ P(\text{Even first and Odd second}) = P(\text{Even first}) \times P(\text{Odd second}) = \frac{8}{17} \times \frac{9}{17} = \frac{72}{289} \]

### Step 6: Determine the value of \( k \)
According to the problem, this probability can be expressed as:
\[ P(\text{Even first and Odd second}) = \frac{k}{289} \]
From our calculation, we have:
\[ \frac{72}{289} = \frac{k}{289} \]
Thus, we find:
\[ k = 72 \]

### Step 7: Find the number of divisors of \( k \)
To find the number of divisors of \( k = 72 \), we first need to find its prime factorization:
\[ 72 = 2^3 \times 3^2 \]
To find the number of divisors, we use the formula:
\[ \text{Number of divisors} = (e_1 + 1)(e_2 + 1) \ldots (e_n + 1) \]
where \( e_i \) are the exponents in the prime factorization.

For \( 72 = 2^3 \times 3^2 \):
- The exponent of 2 is 3, so \( e_1 + 1 = 3 + 1 = 4 \)
- The exponent of 3 is 2, so \( e_2 + 1 = 2 + 1 = 3 \)

Thus, the number of divisors is:
\[ \text{Number of divisors} = 4 \times 3 = 12 \]

### Final Answer
The number of divisors of \( k \) is \( 12 \).

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