Q. Consider the following statements:
I. There exists a natural number which when increased by 50% can have its number of factors unchanged.
II. There exists a natural number which when increased by 150% can have its number of factors unchanged.
Which of the statements given above is/are correct?
(a) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
Correct Answer: (c) Both I and II
UPSC Prelims 2025 CSAT
Explanation :
Statement I (increase by 50% ⇒ factor unchanged)
Let n=2.
• 50% increase: 1.5·2 = 3
• Number of divisors of 2: d(2)=2 (1 and 2)
• Number of divisors of 3: d(3)=2 (1 and 3)
So d(2)=d(3). Hence I is true.
Statement II (increase by 150% ⇒ factor unchanged)
A simple example is n=6.
• 150% increase: 2.5·6 = 15
• d(6)=4 (1, 2, 3, 6)
• d(15)=4 (1, 3, 5, 15)
So d(6)=d(15). Hence II is true.
Since both statements are satisfied by concrete examples, the correct answer is (c) Both I and II.
