Correct Option (3)

Both I and II

Explanation

To determine the number of factors of a natural number, its prime factorization is utilized. If a natural number `n` is expressed as , then the total number of factors of `n` is given by .

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

Increasing a number `n` by means multiplying it by 1.5, or . The new number, `n'`, is .
For to be a natural number, `n` must be even.
Let `n` be represented as , where `K` is not divisible by 2 or 3.
Then, the new number `n'` is . For `n'` to be a natural number, `a` must be at least 1.
The number of factors of `n` is .
The number of factors of `n'` is .
For the number of factors to remain unchanged, we set them equal: .
Solving this equation yields .
This condition implies that for a natural number `n` of the form `2^a * 3^b * K`, if `a = b+1`, its number of factors can remain unchanged. For example, consider `n=2`. Here, `a=1`, `b=0`, and `K=1`, satisfying `a = b+1`. The number of factors of `n=2` is 2 (1, 2). When increased by 50%, `n'` becomes `2 * 1.5 = 3`. The number of factors of `n'=3` is 2 (1, 3). Since the number of factors is unchanged, Statement I is correct.

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

Increasing a number `n` by means multiplying it by 2.5, or . The new number, `n'`, is .
For to be a natural number, `n` must be even.
Let `n` be represented as , where `K` is not divisible by 2 or 5.
Then, the new number `n'` is . For `n'` to be a natural number, `a` must be at least 1.
The number of factors of `n` is .
The number of factors of `n'` is .
For the number of factors to remain unchanged, we set them equal: .
Solving this equation also yields .
This condition implies that for a natural number `n` of the form `2^a * 5^b * K`, if `a = b+1`, its number of factors can remain unchanged. For example, consider `n=2`. Here, `a=1`, `b=0`, and `K=1`, satisfying `a = b+1`. The number of factors of `n=2` is 2 (1, 2). When increased by 150%, `n'` becomes `2 * 2.5 = 5`. The number of factors of `n'=5` is 2 (1, 5). Since the number of factors is unchanged, Statement II is correct.

Both statements are correct as demonstrated by the examples where the number of factors remains unchanged under the specified conditions.

Incorrect Options

Options (1), (2), and (4) are incorrect because both Statement I and Statement II are mathematically valid and can be demonstrated with examples where the number of factors remains unchanged.