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 last digit in the expansion of the number \((54D)^{100}\) is 1. What is the value of the digit D?
Statement-I : D > 5
Statement-II: D is a multiple of 3.
Digit D Value: \((54D)^{100}\) Last Digit is 1
The problem asks us to find the specific value of the digit 'D' in the number \((54D)^{100}\). We are told that the last digit of this number, when expanded, is 1. We also have two separate statements providing clues about 'D'. Our task is to figure out if these clues, individually or together, are enough to determine a unique value for 'D'.
Last Digit of \(D^{100}\) Analysis
The crucial information is that the last digit of \((54D)^{100}\) is 1. The last digit of a power calculation is determined solely by the last digit of the base. Here, the base is \(54D\), so its last digit is \(D\). This means the last digit of \((54D)^{100}\) is the same as the last digit of \(D^{100}\).
We need to find which single digits \(D\) (where \(D\) can range from 0 to 9) result in \(D^{100}\) having a last digit of 1.
Let's explore the patterns of the last digits of powers for each possible digit:
| Digit (D) | Last digit of \(D^{100}\) | Reasoning |
|---|---|---|
| 0 | 0 | \(0^{100}\) always ends in 0. |
| 1 | 1 | \(1^{100}\) always ends in 1. |
| 2 | 6 | Powers of 2 cycle through last digits: 2, 4, 8, 6. Since 100 is a multiple of 4, the cycle repeats fully, ending in 6. So, \(2^{100}\) ends in 6. |
| 3 | 1 | Powers of 3 cycle through last digits: 3, 9, 7, 1. Since 100 is a multiple of 4, the cycle repeats fully, ending in 1. So, \(3^{100}\) ends in 1. |
| 4 | 6 | Powers of 4 cycle through last digits: 4, 6. Since 100 is an even number, \(4^{100}\) ends in 6. |
| 5 | 5 | \(5^{100}\) always ends in 5. |
| 6 | 6 | \(6^{100}\) always ends in 6. |
| 7 | 1 | Powers of 7 cycle through last digits: 7, 9, 3, 1. Since 100 is a multiple of 4, the cycle repeats fully, ending in 1. So, \(7^{100}\) ends in 1. |
| 8 | 6 | Powers of 8 cycle through last digits: 8, 4, 2, 6. Since 100 is a multiple of 4, the cycle repeats fully, ending in 6. So, \(8^{100}\) ends in 6. |
| 9 | 1 | Powers of 9 cycle through last digits: 9, 1. Since 100 is an even number, \(9^{100}\) ends in 1. |
From this analysis, the possible values for the digit \(D\) such that \(D^{100}\) ends in 1 are \(D \in \{1, 3, 7, 9\}\).
Statement I: D > 5 Sufficiency Check
Statement-I says: D > 5
If we only consider Statement I, we look at our possible values \(\{1, 3, 7, 9\}\) and select those that are greater than 5. This gives us \(D=7\) and \(D=9\). Since there are still two possible values for \(D\), Statement I alone is not sufficient to answer the question uniquely.
Statement II: D is a Multiple of 3 Sufficiency Check
Statement-II says: D is a multiple of 3
If we only consider Statement II, we look at our possible values \(\{1, 3, 7, 9\}\) and select those that are multiples of 3. This gives us \(D=3\) and \(D=9\). Since this statement also leaves us with two possible values for \(D\), Statement II alone is not sufficient to answer the question uniquely.
Combining Statements I and II Analysis
Now, let's see if using both statements together helps us find a unique value for \(D\).
- From Statement I, we know \(D\) must be either 7 or 9.
- From Statement II, we know \(D\) must be a multiple of 3.
We need to find a value for \(D\) that satisfies both conditions simultaneously. Looking at the possibilities from Statement I (\(\{7, 9\}\)), we check which one is also a multiple of 3. The digit 9 is a multiple of 3, while 7 is not. Therefore, by combining both statements, we can uniquely determine that \(D=9\).
Since we found a single, specific value for \(D\) by using both statements together, the question can be answered using the information from both statements.
Conclusion on Data Sufficiency
Neither Statement I nor Statement II, when considered individually, provides enough information to determine the value of \(D\). However, when both statements are used in conjunction, they allow us to identify a unique value for \(D\).
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.
Question : Is the triangle equilateral?
Statement-I: \(a^2 + b^2 + c^2 = (ab + bc + ca)\)
Statement-II: \(3a^2 + 3b^2 + 4c^2 = 2ab + 4bc + 4ca\)
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 \(60^\circ\).
