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 ✓ UPSC's answer
- (d) Neither I nor II
Why the answer is (c)
• Statement I: take 2, which has 2 factors (1, 2). Increasing it by 50% gives 3, which also has 2 factors (1, 3). Such a number exists, so Statement I is correct.
• Statement II: take 2 again. Increasing it by 150% gives 2 + 3 = 5, which has 2 factors (1, 5). So Statement II is also correct.
• Both statements are correct — option (c).
Why the other options are wrong
- (a) I only
- This option is incorrect because Statement II is also correct, as demonstrated by the number 2.
- (b) II only
- This option is incorrect because Statement I is also correct, as demonstrated by the number 2.
- (d) Neither I nor II
- This option is incorrect because both statements are correct, as shown by the counterexamples provided.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.