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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?

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    The square root of an even *square* number is even, but not for all of them... ($\sqrt(2)$)2012-10-04
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    You should check to be sure that the problem is not $\rm\:4\:|\:n^2\!\! \iff 2\:|\:n\ \ $2012-10-04
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    Think very carefully. Can you think of a number $n$ which is NOT divisible by 4, but where $n^2$ is even? Just think through some numbers, like 1, 2, 3 and so on. EITHER you'll come to such a number, OR you'll start to gain an understanding of why such numbers mightn't exist. I don't think the problem will take you very long if you adopt this approach.2012-10-04

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