Q. A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: What is the smallest 1-digit number having exactly 4 distinct factors?
Statement I: 2 is one of the factors.
Statement II: 3 is one of the factors.
Which one of the following is correct in respect of the above Question and the Statements?
(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other statement alone.
(b) The Question can be answered by using either Statement alone.
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
(d) The Question can be answered even without using any of the Statements
Correct Answer : a)
UPSC Prelims 2025 CSAT
Explanation :
1. Find all 1-digit numbers with exactly 4 distinct positive factors.
– List 1-digit numbers 1 through 9 and their factor counts:
1 → {1} (1 factor)
2 → {1,2} (2 factors)
3 → {1,3} (2 factors)
4 → {1,2,4} (3 factors)
5 → {1,5} (2 factors)
6 → {1,2,3,6} (4 factors)
7 → {1,7} (2 factors)
8 → {1,2,4,8} (4 factors)
9 → {1,3,9} (3 factors)
– So the candidates are 6 and 8, and the smallest is 6.
2. Use Statement I (“2 is one of the factors”):
– Both 6 and 8 have 2 as a factor.
– We cannot decide between 6 and 8.
– So Statement I alone is insufficient.
3. Use Statement II (“3 is one of the factors”):
– Among 1-digit numbers with 4 factors, only 6 has 3 as a factor.
– This pins down the answer uniquely to 6.
– Statement II alone is sufficient.
Conclusion: Statement II alone suffices, but Statement I does not. Hence option (a).
