Determine whether the following set is finite or infinite:
The set of positive integers greater than $100$.

What is the remainder when the polynomial $P(x) = 1 + x + x^3 + x^9 + x^{27} + x^{81} + x^{243}$ is divided by $x - 1$?

The number of elements in the set $\{ (a, b) : 2a^2 + 3b^2 = 35, a, b \in Z \} $,where $Z$ is the set of all integers,is

If a set $A$ has $n$ elements,then the total number of subsets of $A$ is

If $A = \{x \in \mathbb{R} : x^2 + 5|x| + 6 = 0\}$,then $n(A) = $

In each of the following,determine whether the statement is true or false. If it is true,prove it. If it is false,give a counterexample.
If $A \subset B$ and $B \in C,$ then $A \in C$.