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 \(D\).
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