The sum of all the elements in the set $\{n \in \{1, 2, \ldots, 100\} \mid \text{H.C.F. of } n \text{ and } 2040 \text{ is } 1\}$ is equal to $.....$

In a certain examination,a candidate has to pass in each of the $5$ subjects. The number of ways in which the candidate can fail is:

Let $Z$ be the set of integers. If $A = \{ x \in Z : 2^{(x + 2)(x^2 - 5x + 6)} = 1 \}$ and $B = \{ x \in Z : -3 < 2x - 1 < 9 \}$,then the number of subsets of the set $A \times B$ is:

In a group of $65$ people,$40$ like cricket and $10$ like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

In a school,$20$ teachers teach either Mathematics or Physics. If $12$ teachers teach Mathematics and $4$ teachers teach both subjects,then the number of teachers teaching only Physics is:

$A$ market research group conducted a survey of $1000$ consumers and reported that $720$ consumers like product $A$ and $450$ consumers like product $B$. What is the least number of consumers that must have liked both products?