Is "$n$ is divisible by $4$ if and only if $n^2$ is even" a True Statement? I'm working on some high school geometry homework, and I'm having some trouble with a problem about proofs and counterexamples. The question posses the statement


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*$n$ is divisible by $4$ if and only if $n^2$ is even


and asks if that is a true statement (and to provide a counter example if it is not). My understanding of the statement is that "a prerequisite of divisibility by $4$ is that a number is even when squared." Since the square root of an even number is also even (even $\cdot$ even = even), and the definition of an even number is even divisibility by $2$, the statement can be reduced to "a prerequisite to divisibility by $4$ is divisibility by $2$", which is clearly true. However, I'm concerned that my understanding of the statement is fundamentally flawed. Is the statement true or false, and why?
 A: It is false. Let $n=2$.${}{}{}{}{}{}{}{}$
The sentence has an "if and only if."
So it is claiming that (i) if $n$ is divisible by $4$, then $n^2$ is even (true) and (ii) if $n^2$ is even, then $n$ is divisible by $4$ (false, easy counterexample).
If I claim that Ottawa is the capital of Canada and elephants fly, then I am uttering a falsehood. 
Remark: In general, if you are trying to prove that $A$ if and only if $B$, the first step is to separate the assertion into its two component parts: if $A$ then $B$, and if $B$ then $A$. One part may be correct and the other not, in which case the if and only if assertion is false. Or  both may be true, but the proof for one may be very different from the proof for the other.  
A: You’ve correctly understood half of the statement. A statement of the form $A$ if and only if $B$ means both that $B$ is a prerequisite for $A$ and that $A$ is a prerequisite for $B$. Equivalently, it means that $A$ implies $B$ and $B$ implies $A$.
In you specific case, the statement means that if $n$ is divisible by $4$, then $n^2$ is even, which is true, and that if $n^2$ is even, then $n$ is divisible by $4$, which is not true: just take $n=2$.
However, the following statement is true, and it might be instructive to convince yourself of this:

$n^2$ is divisible by $4$ if and only if $n$ is even.

A: $\rm\begin{eqnarray}{\bf Hint}\ \  prime\ \ p\:|\:nm&\iff&\rm p\:|\:n\ \ or\ \ p\:|\:m \\
\rm hence\ \                prime\ \ p\:|\: n^2 &\iff&\rm p\:|\:n\end{eqnarray} $
Equivalently, $\rm\ mod\ p\!:\ nm\equiv 0\!\iff\! n\equiv 0\ \ or\ \ m\equiv 0,\ \ thus\ \ n^2\equiv 0\!\iff\! n\equiv 0.$
A: I think the appropriate question is:
Is "$n^{2}$ is divisible by 4 if and only if $n$ is even" a true statement?
and the answer is sure.
