An uninitiated student once visited a test prep portal www. Lamamla. in to assess her quantitative ability for SSC CGL. She took a test in which there were total 18 problems each with four option-A, B, C and D. After glancing through the whole test paper she realized that she was not able to crack even a single problems. Then after a while she changed the answers by marking B in each third problem, starting with the third problem. Unsatisfied with her answer, she again changed the answer by marking C in each second problem, starting with the second problem. In her final attempt to guess the answer,she changed the answers by marking D in each ninth problem, starting with the ninth problem. While.changing the answer she moves from the first to the last problem in an orderly way. What are the number of final answer that she marked in terms of A, B, C, and D, respectively?

Options

A. 6, 2, 1, 9
B. 6, 2, 8, 2
C. 2, 4, 6, 6
D. 4, 6, 2, 6

Answer

To solve the problem step by step, we will track the answers marked by the student as she changes her answers according to the given instructions.

### Step-by-Step Solution:

1. **Initial Setup:**
- There are 18 problems, each initially marked as A.
- We can represent the answers for the 18 problems as follows:
\[ \text{Answers} = [A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A, A] \]

2. **Marking B in Each Third Problem:**
- The student marks B in every third problem starting from the third problem. The third problems are 3, 6, 9, 12, 15, and 18.
- Update the answers:
\[ \text{Answers} = [A, A, B, A, A, B, A, A, B, A, A, B, A, A, B, A, A, B] \]

3. **Marking C in Each Second Problem:**
- Next, the student marks C in every second problem starting from the second problem. The second problems are 2, 4, 6, 8, 10, 12, 14, 16, and 18.
- Update the answers:
- For problem 2, change A to C.
- For problem 4, change A to C.
- For problem 6, change B to C.
- For problem 8, change A to C.
- For problem 10, change A to C.
- For problem 12, change B to C.
- For problem 14, change A to C.
- For problem 16, change A to C.
- For problem 18, change B to C.
- The updated answers now look like:
\[ \text{Answers} = [A, C, C, C, A, C, A, C, B, C, A, C, A, C, A, C, A, C] \]

4. **Marking D in Each Ninth Problem:**
- Finally, the student marks D in every ninth problem starting from the ninth problem. The ninth problems are 9 and 18.
- Update the answers:
- For problem 9, change B to D.
- For problem 18, change C to D.
- The final answers are:
\[ \text{Answers} = [A, C, C, C, A, C, A, C, D, C, A, C, A, C, A, C, A, D] \]

5. **Counting Final Answers:**
- Count the occurrences of each option:
- A: 6 times
- B: 0 times
- C: 8 times
- D: 2 times

### Final Answer:
The number of final answers marked in terms of A, B, C, and D, respectively, is:
\[ \text{A: 6, B: 0, C: 8, D: 2} \]