Consider the sets $A = \{ 4 m \mid m$ is a natural number $\}$ and $B = \{ 6 m \mid m$ is a natural number $\}$. (1) For each of the following $\mathbf { L } \sim \mathbf{O}$, choose the correct answer from among (0) $\sim$ (3) below. Let $n$ be a natural number. (i) $n \in A$ is $\mathbf { L }$ for $n$ to be divisible by 2 . (ii) $n \in B$ is $\mathbf { M }$ for $n$ to be divisible by 24 . (iii) $n \in A \cup B$ is $\mathbf { N }$ for $n$ to be divisible by 3 . (iv) $n \in A \cap B$ is $\square\mathbf{O}$ for $n$ to be divisible by 12 . (0) a necessary and sufficient condition (1) a necessary condition but not a sufficient condition (2) a sufficient condition but not a necessary condition (3) neither a necessary condition nor a sufficient condition (2) Let $C = \{ m \mid m$ is a natural number satisfying $1 \leqq m \leqq 100 \}$. The number of elements which belong to $( \bar { A } \cup \bar { B } ) \cap C$ is $\mathbf { P Q }$, and the number of elements which belong to $\bar { A } \cap \bar { B } \cap C$ is $\mathbf { R S }$. Note that $\bar { A }$ and $\bar { B }$ denote the complements of $A$ and $B$, where the universal set is the set of all natural numbers.
Consider the sets $A = \{ 4 m \mid m$ is a natural number $\}$ and $B = \{ 6 m \mid m$ is a natural number $\}$.\\
(1) For each of the following $\mathbf { L } \sim \mathbf{O}$, choose the correct answer from among (0) $\sim$ (3) below.
Let $n$ be a natural number.\\
(i) $n \in A$ is $\mathbf { L }$ for $n$ to be divisible by 2 .\\
(ii) $n \in B$ is $\mathbf { M }$ for $n$ to be divisible by 24 .\\
(iii) $n \in A \cup B$ is $\mathbf { N }$ for $n$ to be divisible by 3 .\\
(iv) $n \in A \cap B$ is $\square\mathbf{O}$ for $n$ to be divisible by 12 .\\
(0) a necessary and sufficient condition\\
(1) a necessary condition but not a sufficient condition\\
(2) a sufficient condition but not a necessary condition\\
(3) neither a necessary condition nor a sufficient condition\\
(2) Let $C = \{ m \mid m$ is a natural number satisfying $1 \leqq m \leqq 100 \}$.
The number of elements which belong to $( \bar { A } \cup \bar { B } ) \cap C$ is $\mathbf { P Q }$, and the number of elements which belong to $\bar { A } \cap \bar { B } \cap C$ is $\mathbf { R S }$. Note that $\bar { A }$ and $\bar { B }$ denote the complements of $A$ and $B$, where the universal set is the set of all natural numbers.