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?
Answer & explanation
Answer: (c) Both I and II
'There exists' needs only one example. The number 2 becomes 3 after a 50% increase and 5 after a 150% increase; all three are primes with exactly two factors, so both statements are correct.
- Statement I: 2 has 2 factors. Increased by 50%, it becomes 3, which also has 2 factors. So I is correct.
- Statement II: 2 increased by 150% becomes 2 × 2.5 = 5, which also has 2 factors. So II is correct.
- Both statements are correct.
- Check: other examples exist too, e.g. 10 → 15 (4 factors each) for I and 6 → 15 (4 factors each) for II.
- ✓ I 2 → 3: both are primes with exactly 2 factors.
- ✓ II 2 → 5: both are primes with exactly 2 factors.
Remember · 'There exists' statements need just one example — try small primes first.
Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·