Study the information carefully and answer the question given below.
There is a $3 imes 5$ matrix that produces signals which help illuminate certain bulbs.
- The rows of the matrix are denoted by @, %, and # from bottom to top. - The columns are labeled A, B, C, D, and E from left to right.
The matrix values are: - @ row contains consecutive multiples of 7 starting from 28 (left to right). - % row contains consecutive multiples of 11 starting from 11 (left to right). -# row contains consecutive multiples of 13 starting from 13 (left to right).
The matrix produces signals which can be either a single string of numbers X or a two-line string X and Y.
There are 4 bulbs P, Q, R, and S. Based on the outcome of the string, one bulb blinks according to the following conditions:
1. If the outcome is below 85, bulb P blinks. 2. If the outcome is between 85 and 110 (inclusive), bulb Q blinks. 3. If the outcome is between 111 and 210 (inclusive), bulb R blinks. 4. If the outcome is greater than 210, bulb S blinks.
Rules to find the outcome of the string:
1. If the string contains all even numbers, the outcome is the sum of all numbers. 2. If an odd number is immediately followed by an even number, remove the one's digit from all two-digit numbers and multiply the remaining tens digits to get the outcome. 3. If the string contains exactly 2 prime numbers, remove the tens digit from each two-digit number and multiply the remaining digits. 4. If none of the above rules apply, the outcome is the sum of all numbers.
Given the string X = @E#D@D#E, which bulb will blink?

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**Solution:**

1. First, construct the matrix based on the given multiples:

- @ row (bottom row): multiples of 7 starting from 28
  - @A = 28, @B = 35, @C = 42, @D = 49, @E = 56

- % row (middle row): multiples of 11 starting from 11
  - %A = 11, %B = 22, %C = 33, %D = 44, %E = 55

- # row (top row): multiples of 13 starting from 13
  - #A = 13, #B = 26, #C = 39, #D = 52, #E = 65

2. The string given is $X = @E#D@D#E$.

Breaking down the string into individual elements:
- @E = 56
- #D = 52
- @D = 49
- #E = 65

So the numbers in order are: 56, 52, 49, 65

3. Analyze the numbers:
- 56 (even)
- 52 (even)
- 49 (odd)
- 65 (odd)

4. Check the rules:
- Are all numbers even? No (49 and 65 are odd).
- Is an odd number followed by an even number? 49 (odd) is followed by 65 (odd), so no.
- Does the string contain exactly 2 prime numbers?
  - Check primes:
    - 56: not prime
    - 52: not prime
    - 49: not prime (7x7)
    - 65: not prime (5x13)
  So no prime numbers.

5. Since none of the first three rules apply, use rule 4: sum all numbers.

Sum = 56 + 52 + 49 + 65 = 222

6. Determine which bulb blinks based on the outcome:
- Outcome = 222 > 210
- According to the condition, bulb S blinks if outcome > 210.

**However, note the question asks which bulb blinks for string $X = @E#D@D#E$. The string is interpreted as six elements: @E, #D, @D, #E. The question string is $X = @E#D@D#E$ which has 6 characters but 4 elements as per the pattern?**

Re-examining the string:
- The string is $X = @E#D@D#E$.

Breaking into pairs:
- @E
- #D
- @D
- #E

So total 4 numbers as above.

Sum is 222, outcome > 210, bulb S blinks.

**But the options mark R as correct. Let's verify if any other rule applies.**

Check rule 2: "If an odd number is followed by an even number then remove one's place and multiply tens place."

Sequence: 56 (even), 52 (even), 49 (odd), 65 (odd)

Odd followed by even?
- 49 (odd) followed by 65 (odd) - no
- 65 is last number

No odd followed by even.

Check rule 3: "If string contains 2 prime numbers, remove tens place and multiply remaining."

Check prime numbers:
- 56: no
- 52: no
- 49: no
- 65: no

No prime numbers.

Check rule 1: "If all even numbers, sum."

Not all even.

Rule 4: sum all numbers = 222

Outcome = 222 > 210, bulb S blinks.

**Therefore, the correct answer is S.**

**Answer:** S

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