If `A = {9, 10, 11, 12, 13}` and let `f : A rarr N` is defined by `f(x)` = highest prime factor of x. then the number of distinct elements in the image of f is :
`"(a) 6 (b)7 (c) 8 (d) 4 "`

Options

A. `6`
B. `7`
C. `8`
D. `4`

Explanation

To solve the problem, we need to determine the highest prime factor for each element in the set \( A = \{9, 10, 11, 12, 13\} \) and then find the number of distinct values in the image of the function \( f \).

### Step-by-Step Solution:

1. **Identify the elements of set A**:
\[ A = \{9, 10, 11, 12, 13\} \]

2. **Calculate \( f(9) \)**:
- The factors of 9 are \( 1, 3, 9 \).
- The prime factors are \( 3 \).
- Therefore, \( f(9) = 3 \).

3. **Calculate \( f(10) \)**:
- The factors of 10 are \( 1, 2, 5, 10 \).
- The prime factors are \( 2, 5 \).
- The highest prime factor is \( 5 \).
- Therefore, \( f(10) = 5 \).

4. **Calculate \( f(11) \)**:
- The factors of 11 are \( 1, 11 \).
- The prime factor is \( 11 \).
- Therefore, \( f(11) = 11 \).

5. **Calculate \( f(12) \)**:
- The factors of 12 are \( 1, 2, 3, 4, 6, 12 \).
- The prime factors are \( 2, 3 \).
- The highest prime factor is \( 3 \).
- Therefore, \( f(12) = 3 \).

6. **Calculate \( f(13) \)**:
- The factors of 13 are \( 1, 13 \).
- The prime factor is \( 13 \).
- Therefore, \( f(13) = 13 \).

7. **List the values of \( f(x) \)**:
- From the calculations, we have:
- \( f(9) = 3 \)
- \( f(10) = 5 \)
- \( f(11) = 11 \)
- \( f(12) = 3 \)
- \( f(13) = 13 \)

8. **Determine the distinct elements in the image of \( f \)**:
- The values obtained are \( \{3, 5, 11, 13\} \).
- The distinct elements are \( 3, 5, 11, 13 \).

9. **Count the distinct elements**:
- There are \( 4 \) distinct elements in the image of \( f \).

### Final Answer:
The number of distinct elements in the image of \( f \) is \( \boxed{4} \).