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 is correct: Consider the natural number 1, which has 1 factor. Increasing it by 50% gives 1.5, which is not a natural number, so we must look for an integer result. Let the number be $n$. $1.5n$ must be an integer, so $n$ must be even. Let $n=2$. Factors of 2 are {1, 2} (count=2). $1.5 \times 2 = 3$. Factors of 3 are {1, 3} (count=2). The number of factors remains unchanged.

• Statement II is correct: Consider the natural number 2. Increasing it by 150% means adding $1.5 \times 2 = 3$, resulting in $2+3=5$. Factors of 2 are {1, 2} (count=2). Factors of 5 are {1, 5} (count=2). The number of factors remains unchanged.

• Since both statements describe valid scenarios where a natural number exists satisfying the condition, both are correct.

• Therefore, the option stating that both I and II are correct is the right answer.

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.