A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: The product of a natural number N and the number M written by the same digits of N in the reverse order is 252. What is the number N? Statement-I: N+ M = 33 Statement-II: N > M Which one of the following is correct in respect of the above Question and the Statements?

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Question

CDS - 2025

A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
The product of a natural number N and the number M written by the same digits of N in the reverse order is 252. What is the number N?
Statement-I: N+ M = 33
Statement-II: N > M

Which one of the following is correct in respect of the above Question and the Statements?

This question was previously asked in

CDS 2 2025 Maths Question Paper (14-Sep-2025)

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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

The correct answer is

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

Analyzing the Data Sufficiency Problem

The question asks us to find a natural number N such that the product of N and M (where M is the number formed by reversing the digits of N) is 252. We are given two statements and need to determine if they are sufficient to answer the question.

The core condition is: \(N \times M = 252\) where M is the reverse of N.

Evaluating Statement-I

Statement-I provides an additional condition: \(N + M = 33\)

We now have a system of two equations with two variables, N and M:

\(N \times M = 252\)
\(N + M = 33\)

We can think of N and M as the roots of a quadratic equation of the form:

\(x^2 - (\text{Sum of roots})x + (\text{Product of roots}) = 0\)

Substituting the values from our equations:

\(x^2 - (33)x + (252) = 0\)

To solve this quadratic equation, we can factorize it. We need two numbers that multiply to 252 and add up to 33. Let's list factors of 252:

1, 252	2, 126	3, 84	4, 63	6, 42	7, 36	9, 28	12, 21	14, 18
Sum: 253	Sum: 128	Sum: 87	Sum: 67	Sum: 48	Sum: 43	Sum: 37	Sum: 33	Sum: 32

The pair (12, 21) adds up to 33.

So, the quadratic equation becomes:

\((x - 12)(x - 21) = 0\)

The possible values for x (which represent N and M) are 12 and 21.

This means the possible pairs for (N, M) are (12, 21) or (21, 12).

If N = 12, then M = 21. M is the reverse of N. \(12 \times 21 = 252\). \(12 + 21 = 33\). This fits.
If N = 21, then M = 12. M is the reverse of N. \(21 \times 12 = 252\). \(21 + 12 = 33\). This also fits.

Since there are two possible values for N (12 and 21), Statement-I alone is not sufficient to answer the question.

Evaluating Statement-II

Statement-II provides an additional condition: \(N > M\)

We know from the original question that \(N \times M = 252\) and M is the reverse of N. The possible pairs (N, M) that satisfy this are (12, 21) and (21, 12).

Now, let's apply the condition \(N > M\) to these pairs:

Case 1: (N, M) = (12, 21). Is \(12 > 21\)? No, this is false.
Case 2: (N, M) = (21, 12). Is \(21 > 12\)? Yes, this is true.

Only the pair (21, 12) satisfies the condition \(N > M\). Therefore, N must be 21.

Statement-II alone is sufficient to uniquely determine the value of N.

Conclusion on Sufficiency

Based on the analysis:

Statement-I alone is not sufficient.
Statement-II alone is sufficient.

This situation corresponds to the option where the question can be answered using one statement alone, but not the other.

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