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

Find the pairs of equal sets,if any,give reasons:
$A = \{ 0 \}$
$B = \{ x : x > 15 \text{ and } x < 5 \}$
$C = \{ x : x - 5 = 0 \}$
$D = \{ x : x^2 = 25 \}$
$E = \{ x : x \text{ is an integral positive root of the equation } x^2 - 2x - 15 = 0 \}$

Let $A$ and $B$ be two non-empty subsets of a set $X$ such that $A$ is not a subset of $B$. Then:

State whether the following set is finite or infinite:
The set of numbers which are multiples of $5$.

In the set-builder notation (rule method),the null set is represented by:

Write the following set in the set-builder form: $\{ 1, 4, 9, \ldots, 100 \}$