"A network of voluntary associations stands as a 'buffer' between the relatively powerless individual and the potentially powerful State."

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?

"A network of voluntary associations stands as a 'buffer' between the relatively powerless individual and the potentially powerful State."

What is the remainder when $9^3+9^4+9^5+9^6+\ldots+9^{100}$ is divided by 6 ?

Trust stands commonly defined as being vulnerable to others. Entrepreneurship implies trust in others and willingness to expose oneself to betrayal. Trust in expert systems is the essence of globalising behaviour; trust itself emerges as a supercommodity in the social market and defines the characteristics of goods and services in a global market. Trusting conduct also means holding others in good esteem, and an optimism that they are, or will be, competent in certain respects.

In a T20 cricket match, three players X, Y and Z scored a total of 37 runs. The ratio of number of runs scored by $\mathbf{X}$ to the number of runs scored by $\mathbf{Y}$ is equal to ratio of number of runs scored by $Y$ to number of runs scored by $\mathbf{Z}$.

$$ \begin{aligned} & \text { Value-I = Runs scored by X } \\ & \text { Value-II = Runs scored by Y } \end{aligned} $$

Value-III $=$ Runs scored by $\mathbf{Z}$

Which one of the following is correct?

Trust stands commonly defined as being vulnerable to others. Entrepreneurship implies trust in others and willingness to expose oneself to betrayal. Trust in expert systems is the essence of globalising behaviour; trust itself emerges as a supercommodity in the social market and defines the characteristics of goods and services in a global market. Trusting conduct also means holding others in good esteem, and an optimism that they are, or will be, competent in certain respects.

Let $p+q=10$, where $p, q$ are integers.

Value-I $=$ Maximum value of $\mathbf{p} \times \mathbf{q}$ when $\mathbf{p}, \mathbf{q}$ are positive integers.

Value-II $=$ Maximum value of $p \times q$ when $p \geq-6, q \geq-4$.

Which one of the following is correct?