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
- $n$ 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?