Consider the following statements : There exists a natural number which I. when increased by 50% can have its number of factors unchanged. There exists a natural number which II. when increased by 150% can have its number of factors unchanged. Which of the statements given above is/are correct ?

To determine if the statements are correct, we look for natural numbers that satisfy the conditions.

Statement I: A natural number increased by 50 percent is the same as multiplying it by 1.5 or 3/2. For the result to remain a natural number, the original number must be even. Let us take the number 6. The factors of 6 are 1, 2, 3, and 6, totaling 4 factors. If we increase 6 by 50 percent, we get 9. The factors of 9 are 1, 3, and 9, totaling 3 factors. However, the statement asks if the number of factors can remain unchanged. Let us take the number 12. Its factors are 1, 2, 3, 4, 6, and 12, totaling 6 factors. If we increase 12 by 50 percent, we get 18. The factors of 18 are 1, 2, 3, 6, 9, and 18, which also totals 6 factors. Thus, statement I is correct.

Statement II: A natural number increased by 150 percent is the same as multiplying it by 2.5 or 5/2. For the result to remain a natural number, the original number must be even. Let us take the number 2. Its factors are 1 and 2, totaling 2 factors. If we increase 2 by 150 percent, we get 5. The factors of 5 are 1 and 5, which also totals 2 factors. Thus, statement II is correct.

Since both statements are correct, the answer is C.