Question Details

A question is given followed by two Statements I and II.
ConsidertheQuestionandtheStatementsandmark the correct option.
Question:
Let P, Q, R, S be distinct non-zero digits. If $P P \times P Q = R R S S$ ,where $P \leq 3$ and $Q \leq 4$ ,thenwhatisQequalto?
Statement I: R = 1.
Statement II: S = 2.
Whichoneofthefollowingiscorrectinrespectofthe above Question and the Statements?

Options

The Question can be answered by using one of the Statements alone, but cannot be answered using the other statement alone.

The Question can be answered by using either Statement alone.

The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

The Question can be answered even without using any of the Statements.

Show Answer

Correct Answer :

Option D

The Question can be answered even without using any of the Statements.

Solution :

The correct option is The Question can be answered even without using any of the Statements.

Let us analyze the problem step-by-step using the given mathematical constraints.

1. Given Information:
• $P$ , $Q$ , $R$ , $S$ are distinct non-zero digits (i.e., elements of {1, 2, 3, 4, 5, 6, 7, 8, 9}).
• The equation is: $P P \times P Q = R R S S$
• $P \leq 3$
• $Q \leq 4$

2. Expressing Numbers algebraically:
• $P P = 11 \times P$
• $R R S S = 1100 R + 11 S = 11 \times (100 R + S)$

Dividing both sides of the main equation by 11 gives:

$P \times P Q = 100 R + S$

3. Testing values of P ( $P \leq 3$ and non-zero):
Since $P$ is a non-zero digit and $P \leq 3$ , $P$ can be 1, 2, or 3.

• If $P = 1$ :
$P P \times P Q = 11 \times 1 Q$ .
Since $Q \leq 4$ , the maximum product is $11 \times 14 = 154$ , which is not a 4-digit number $R R S S$ . Thus, $P \neq 1$ .

• If $P = 2$ :
$P P \times P Q = 22 \times 2 Q$ .
Since $Q \leq 4$ , the maximum product is $22 \times 24 = 528$ , which is also not a 4-digit number. Thus, $P \neq 2$ .

• If $P = 3$ :
$P P = 33$ , and $P Q \in {31, 32, 33, 34}$ since $Q \leq 4$ and non-zero.
Let us test the possible values of $Q$ :
- If $Q = 1$ : $33 \times 31 = 1023$ (Not in the form $R R S S$ )
- If $Q = 2$ : $33 \times 32 = 1056$ (Not in the form $R R S S$ )
- If $Q = 3$ : $Q$ cannot be 3 because $P, Q, R, S$ must be distinct digits, so $Q \neq P$ .
- If $Q = 4$ : $33 \times 34 = 1122$ .

Here, $1122$ fits the pattern $R R S S$ where $R = 1$ and $S = 2$ .
The digits involved are $P = 3$ , $Q = 4$ , $R = 1$ , and $S = 2$ , which are all distinct non-zero digits.

Conclusion:
We uniquely found $Q = 4$ directly from the question's main constraints without needing any additional information from Statement I or Statement II.
Therefore, the Question can be answered even without using any of the Statements.