Using the words "necessary and sufficient",rewrite the statement "The integer $n$ is odd if and only if $n^{2}$ is odd". Also,check whether the statement is true.

(N/A) The statement "The integer $n$ is odd if and only if $n^{2}$ is odd" can be rewritten as: "The condition that the integer $n$ is odd is necessary and sufficient for $n^{2}$ to be odd."
Let $p$ be the statement: "The integer $n$ is odd."
Let $q$ be the statement: "$n^{2}$ is odd."
To check the validity of $p \iff q$,we examine both implications:
$1$. If $p$ is true,then $n = 2k + 1$ for some integer $k$. Then $n^{2} = (2k + 1)^{2} = 4k^{2} + 4k + 1 = 2(2k^{2} + 2k) + 1$,which is odd. Thus,$p \implies q$ is true.
$2$. If $q$ is true,we use the contrapositive: if $n$ is even,then $n = 2k$. Then $n^{2} = (2k)^{2} = 4k^{2} = 2(2k^{2})$,which is even. Since the contrapositive is true,$q \implies p$ is true.
Since both $p \implies q$ and $q \implies p$ are true,the original statement is true.