A Question is given followed by two Statements I and II. Consider the Question and the Statements. Which one of the following is correct in respect of the above 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
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.
Similar Questions
- Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q.
Question : Is \(P > Q\)?
Statement-I: \(x : y = 2 : 1\)
Statement-II: The angle between the two adjacent sides of the parallelogram is \(60^\circ\). - A circle touches all the four sides AB, BC, CD, DA of a quadrilateral ABCD.
Question : What is the perimeter of the quadrilateral?
Statement-I: \(AB + DC = 10\) cm
Statement-II: \(AD + BC = 10\) cm - P, Q, R, S are the mid-points of sides AB, BC, CD, DA respectively of a quadrilateral ABCD.
Question : What is the difference in the area of the quadrilateral ABCD and the area of the quadrilateral PQRS?
Statement-I: Area of the quadrilateral ABCD is 100 square unit.
Statement-II: Area of the quadrilateral PQRS is 50 square unit. - Question : What is the ratio of the lengths of diagonals of a rhombus?
Statement-I: One diagonal of the rhombus is equal to its side.
Statement-II: The longer diagonal of the rhombus is equal to \(\sqrt{3}\) times its side. - The chord of a circle of radius R touches at a point on the circumference of a concentric circle of radius r. The length of the chord is 24 units.
Question : What are the values of r and R?
Statement-I: r is an integer.
Statement-II: R is an integer. - Let a, b, c and d be positive integers.
Question : Which one of a, b, c, d is closest to the product abcd?
Statement-I: \(a > b > c\)
Statement-II: c is not the smallest. - Let \(mn = k\), where m and n are prime numbers and k is an even number.
Question : What is the value of \(mn - n + 1\)?
Statement-I: \(m > n\)
Statement-II: One of the numbers is 2. - Question : If p is a positive integer, then what is the remainder when $p^n$ is divided by $p + 1$?
Statement-I: n is even.
Statement-II: p is even. Question : Is xy positive?
Statement-I: \(x = \sqrt[3]{-0.19683}\)
Statement-II: $y = \(\sqrt[3]{x} \)- Let a, b and c be the sides of a triangle ABC.
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Important Questions from Data Sufficiency
- In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Q. Harsh, Ketan, Mayank, Rohit, and Sidharth, each have different height. Who is the shortest?
(I) Mayank is taller than Harsh but shorter than Ketan.
(II) Rohit is taller than Harsh but shorter than Mayank. - In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Q. Kunal, Mohit, Rohan, Sandeep, and Tarun - each have different age. Who is the oldest?
(I) Mohit is older than Kunal but younger than Sandeep.
(II) Tarun is younger than Sandeep but older than Mohit. - In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Question:
Among 5 trees, L, M, N, T and V each having a different height, which tree is the shortest?
Statements:
(I) N is shorter than only three trees.
(II) N is taller than M but shorter than T. Below is given a pair of questions, which are required to be answered, and a triad of statements. Which group of statements is sufficient to answer both the questions?
Question 1: Who is sitting opposite to Q?
Question 2: Who is sitting to the right of Q?
Statement 1 : P and U are sitting on left and right sides of T, respectively
Statement 2: P, Q, R, S, T and U are sitting around a round table with six chairs
Statement 3: Q is NOT sitting on the immediate left side of R
- Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q.
Question : Is \(P > Q\)?
Statement-I: \(x : y = 2 : 1\)
Statement-II: The angle between the two adjacent sides of the parallelogram is \(60^\circ\).
