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?

The correct option is Both I and II.

Let us analyze both statements step-by-step to determine their correctness.

Analysis of Statement I:

Statement I claims: "There exists a natural number which when increased by 50% can have its number of factors unchanged."

Increasing a number

$n$

by 50% means multiplying

$n$

by

$1.5=\frac{3}{2}$

.

Let

$n=6$

.

The number of factors of 6 can be found by listing them: 1, 2, 3, and 6. So, 6 has 4 factors.

Now, increase 6 by 50%:

$6+50\%\text{of}6=6+3=9$

The factors of 9 are: 1, 3, and 9. Thus, 9 has 3 factors. (Number of factors changed here).

Let us try another natural number, for example,

$n=10$

.

The factors of 10 are: 1, 2, 5, and 10 (total 4 factors).

Now, increase 10 by 50%:

$10\times \frac{3}{2}=15$

The factors of 15 are: 1, 3, 5, and 15 (total 4 factors).

Since 10 has 4 factors and 15 also has 4 factors, the number of factors remains unchanged.

Therefore, Statement I is correct.

Analysis of Statement II:

Statement II claims: "There exists a natural number which when increased by 150% can have its number of factors unchanged."

Increasing a number

$n$

by 150% means multiplying

$n$

by

$1+1.5=2.5=\frac{5}{2}$

.

Let us test

$n=14$

.

The factors of 14 are: 1, 2, 7, and 14 (total 4 factors).

Now, increase 14 by 150%:

$14\times \frac{5}{2}=35$

The factors of 35 are: 1, 5, 7, and 35 (total 4 factors).

Since 14 has 4 factors and 35 also has 4 factors, the number of factors remains unchanged.

Therefore, Statement II is also correct.

Conclusion: Since both statements are true, the correct answer is Both I and II.