Which one of the following is correct in respect of the Statements given below?

Statement 1:  $$2^{10}$$​is the smallest natural number having exactly 11 factors. ​

Statement 2: $$2^{11}$$​ is the smallest natural number having exactly 12 factors. ​

Statement 3: $$2^{12}$$​ is the smallest natural number having exactly 13 factors.

Correct option is D

Given:

Statement 1: $$2^{10}$$ is the smallest natural number having exactly 11 factors.

Statement 2:  is the smallest natural number having exactly 12 factors.$$2^{11}$$​

Statement 3: $$2^{12}$$​is the smallest natural number having exactly 13 factors.

Solution:

Check Statement 1

For a number to have 11 factors, the exponent of its prime factor must be 10 (since 10 + 1 = 11).

The smallest such number is 2¹⁰ = 1024.

Statement 1 is correct.

Check Statement 2:

For a number to have 12 factors, the exponents can be arranged as (11), (5, 1), or (3, 2).
The possible numbers are:

​$$2^{11} = 2048$$​​

​$$2^5 \times 3^1 = 96$$​​

​$$2^3 \times 3^2 = 72$$​​

The smallest number with 12 factors is 72, not $$2^{11}$$​.
Statement 2 is incorrect.

Check Statement 3:
For a number to have 13 factors, the exponent of its prime factor must be 12 (since 12 + 1 = 13).
The smallest such number is $$2^{12}.$$​​
Statement 3 is correct.

​Statements 1 and 3 are correct, while statement 2 is incorrect.​

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