A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question: Statement I: 2 is one of the factors. Which one of the following is correct in respect of the above Question and the Statements?
What is the smallest 1-digit number having exactly 4 distinct factors?
Statement II: 3 is one of the factors.
The Question can be answered even without using any of the Statements.
Finding the Smallest 1-Digit Number with Exactly 4 Distinct Factors
The question asks us to find the smallest 1-digit number that has exactly 4 distinct factors. We are also given two statements about this number and asked which statement(s) are needed to answer the question.
Analyzing the Question
First, let's understand what a 1-digit number is. These are the integers from 1 to 9.
Next, we need to understand what distinct factors are. Factors of a number are integers that divide the number evenly. Distinct factors means counting each unique factor only once.
We need to find the 1-digit number that has precisely 4 distinct factors, and among those, identify the smallest one.
Finding Factors of 1-Digit Numbers
Let's list the 1-digit numbers and their factors:
| Number | Factors | Number of Distinct Factors |
|---|---|---|
| 1 | {1} | 1 |
| 2 | {1, 2} | 2 |
| 3 | {1, 3} | 2 |
| 4 | {1, 2, 4} | 3 |
| 5 | {1, 5} | 2 |
| 6 | {1, 2, 3, 6} | 4 |
| 7 | {1, 7} | 2 |
| 8 | {1, 2, 4, 8} | 4 |
| 9 | {1, 3, 9} | 3 |
Identifying Numbers with Exactly 4 Distinct Factors
From the table, the 1-digit numbers that have exactly 4 distinct factors are 6 and 8.
Finding the Smallest Number
We need the smallest of these numbers. Comparing 6 and 8, the smallest number is 6.
So, the smallest 1-digit number having exactly 4 distinct factors is 6.
We were able to determine this number (6) by simply analyzing the definition of 1-digit numbers and factors, without needing any additional information.
Analyzing the Statements
Now let's look at the statements:
- Statement I: 2 is one of the factors. The number we found is 6. Factors of 6 are {1, 2, 3, 6}. 2 is indeed a factor of 6. If we only used Statement I, we would know the number has 4 factors and 2 is one of them. This could be 6 or 8 among 1-digit numbers with 4 factors. So Statement I alone is not sufficient to pinpoint 6.
- Statement II: 3 is one of the factors. The number we found is 6. Factors of 6 are {1, 2, 3, 6}. 3 is indeed a factor of 6. If we only used Statement II, we would know the number has 4 factors and 3 is one of them. Among 1-digit numbers with 4 factors (6 and 8), only 6 has 3 as a factor. So Statement II alone *would* point to 6.
However, the core task was to find the smallest 1-digit number with exactly 4 distinct factors. We did this by systematically checking the numbers from 1 to 9, which revealed that 6 is the smallest such number. The statements provide properties of this number (6), but were not necessary to find the number itself based on the original criteria (smallest 1-digit number with exactly 4 factors).
Conclusion based on Statements
Since we could find the number (6) directly from the question's criteria (smallest 1-digit number with exactly 4 distinct factors) without referring to Statement I or Statement II, the question can be answered even without using any of the statements.
Final Answer Derivation
By examining all 1-digit numbers and their factors, we identified that 6 is the smallest among them with exactly 4 distinct factors. This process did not require using Statement I or Statement II. Therefore, the question can be answered independently of the given statements.
Revision Table: Smallest 1-Digit Number Factors
| Number | Factors | # Factors | Exactly 4 Factors? |
|---|---|---|---|
| 1 | {1} | 1 | No |
| 2 | {1, 2} | 2 | No |
| 3 | {1, 3} | 2 | No |
| 4 | {1, 2, 4} | 3 | No |
| 5 | {1, 5} | 2 | No |
| 6 | {1, 2, 3, 6} | 4 | Yes |
| 7 | {1, 7} | 2 | No |
| 8 | {1, 2, 4, 8} | 4 | Yes |
| 9 | {1, 3, 9} | 3 | No |
From the table, numbers 6 and 8 have exactly 4 factors. The smallest is 6.
Additional Information: Number of Factors
The number of factors a positive integer has is related to its prime factorization. If a number $\text{N}$ can be written as $\text{N} = \text{p}_1^{\text{a}_1} \times \text{p}_2^{\text{a}_2} \times \dots \times \text{p}_{\text{k}}^{\text{a}_{\text{k}}$, where $\text{p}_1, \text{p}_2, \dots, \text{p}_{\text{k}}$ are distinct prime numbers and $\text{a}_1, \text{a}_2, \dots, \text{a}_{\text{k}}$ are positive integers, then the total number of distinct factors of $\text{N}$ is given by the product of one more than each exponent:
Number of factors $= (\text{a}_1 + 1)(\text{a}_2 + 1)\dots(\text{a}_{\text{k}} + 1)$.
For example:
- The number 6 has the prime factorization $2^1 \times 3^1$. The number of factors is $(1+1)(1+1) = 2 \times 2 = 4$.
- The number 8 has the prime factorization $2^3$. The number of factors is $(3+1) = 4$.
This confirms that 6 and 8 are indeed 1-digit numbers with exactly 4 distinct factors.
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