59. A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question: Let P, Q, R, S be distinct non-zero digits. If PP × PQ = RRSS, where P ≤ 3 and Q ≤ 4, then what is Q equal to? Statement I: R = 1. Statement II: S = 2. Which one of the following is correct in respect of the above Question and the Statements?

1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other statement alone.
2. B. The Question can be answered by using either Statement alone.
3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
4. D. The Question can be answered even without using any of the Statements.

Answer: D

Explanation

We are given a multiplication puzzle: PP × PQ = RRSS, where P, Q, R, S are distinct non-zero digits, P ≤ 3, and Q ≤ 4.
PP can be written as 11P. PQ can be written as 10P + Q. RRSS can be written as 1100R + 11S = 11(100R + S).
So, 11P × (10P + Q) = 11(100R + S), which simplifies to P × (10P + Q) = 100R + S.

Let’s test possible values for P (1, 2, 3) and Q (1, 2, 3, 4), ensuring P and Q are distinct:
* If P = 1: Q can be 2, 3, 4. The product 11 × (10+Q) would be 11×12=132, 11×13=143, 11×14=154. These are 3-digit numbers, not 4-digit RRSS. So P cannot be 1.
* If P = 2: Q can be 1, 3, 4. The product 22 × (20+Q) would be 22×21=462, 22×23=506, 22×24=528. These are 3-digit numbers, not 4-digit RRSS. So P cannot be 2.
* If P = 3: Q can be 1, 2, 4 (Q cannot be 3 as digits must be distinct).
* If Q = 1: 33 × 31 = 1023. This is not of the form RRSS (e.g., 1122, 2233). Also, P=3, Q=1, R=1, S=2. R=Q, not distinct.
* If Q = 2: 33 × 32 = 1056. Not of the form RRSS.
* If Q = 4: 33 × 34 = 1122. This is of the form RRSS, where R=1 and S=2. Let’s check distinct non-zero digits: P=3, Q=4, R=1, S=2. All are distinct and non-zero. This is a valid solution.

From this, we uniquely determine Q = 4. We found the value of Q without using Statement I or Statement II. Therefore, the question can be answered even without using any of the statements. This is a challenging number puzzle combined with data sufficiency.