54. 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?

1. A. I only
2. B. II only
3. C. Both I and II
4. D. Neither I nor II

Answer: C

Explanation

Statement I: There exists a natural number which when increased by 50% can have its number of factors unchanged.
Let the natural number be N. Increased by 50%, it becomes N * (1 + 50/100) = N * (3/2).
Consider N = 2. The factors of 2 are (1, 2), so it has 2 factors. When increased by 50%, N becomes 2 * (3/2) = 3. The factors of 3 are (1, 3), so it also has 2 factors. Since the number of factors remains unchanged, Statement I is correct.

Statement II: There exists a natural number which when increased by 150% can have its number of factors unchanged.
Let the natural number be N. Increased by 150%, it becomes N * (1 + 150/100) = N * (5/2).
Consider N = 2. The factors of 2 are (1, 2), so it has 2 factors. When increased by 150%, N becomes 2 * (5/2) = 5. The factors of 5 are (1, 5), so it also has 2 factors. Since the number of factors remains unchanged, Statement II is correct.

Since both statements are correct, option (c) is the answer. This question tests the understanding of factors of numbers and percentage calculations, requiring a bit of trial and error or number sense.