Let $S = \{0, 1, 5, 4, 7\}$. Then the number of all subsets of $S$ is:

Let the set $C = \{(x, y) \mid x^2 - 2^y = 2023, x, y \in \mathbb{N}\}$. Then $\sum_{(x, y) \in C} (x + y)$ is equal to

The set $\{n(n+1)(2n+1) : n \in \mathbb{Z}\}$ is a subset of:

Set $A$ has $m$ elements and Set $B$ has $n$ elements. If the total number of subsets of $A$ is $112$ more than the total number of subsets of $B$,then the value of $m \times n$ is

The number of subgroups of the group $(Z_{5}, +_{5})$ is

If $aN = \{ ax : x \in N \}$ and $bN \cap cN = dN$,where $b, c \in N$ are relatively prime,then