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I can see clearly why $p$ in an even number implies that $p^2$ is even. But I don't understand completely why the opposite statement is also true, can someone show me why? I can start with $$p^2 = 2a$$ but if I'll take the square root of both sides it'll lead to nothing.

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    Hint: $(2n+1)^2=4n^2+4n+1$ – 2017-02-05
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    @lulu it says that a square of an odd number is also odd, I've used it to prove the same statement about an even number, but I can't really tell how can it help me to prove the reversed statement – 2017-02-05
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    @Ozk I think the problem is asking you to use contraposition, using lulu's comment. See my answer. – 2017-02-05
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    Not sure I follow. If you know that $(2n+1)^2$ is always odd, then clearly the square root of an even number can not be odd. – 2017-02-05

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Update: You want to prove:

$$\text{$p^2$ even} \implies \text{$p$ even}$$

which is equivalent by contraposition to:

$$\lnot (\text{$p$ even}) \implies \lnot(\text{$p^2$ even})$$

ie.

$$\text{$p$ odd} \implies \text{$p^2$ odd}$$

This can be proved using the definition by writing $p=2k+1$ and computing $p^2$.

If you don't want to prove it manually, it can be seen also as a direct consequence of Euclid's lemma, namely:

If $q$ divides $ab$, where $q$ is prime and $a,b$ are integers, then $q$ divides $a$ or $q$ divides $b$.

Take $q=2$, $a=b=p$.

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    But I want to prove it manually – 2017-02-05
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    @Ozk: I updated my answer. I think there was previously an answer explaining this, I don't know why it was deleted. – 2017-02-05
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    What's the meaning of "ie." ? – 2017-02-06
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    The manual proof is only exist in an indirect way? – 2017-02-06
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    @Ozk: http://dictionary.cambridge.org/dictionary/english/i-e – 2017-02-06
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    @Ozk: If you want to prove it without using contraposition, you'll need some "structural result" about prime numbers, such as Euclid's lemma (or more advanced result e.g. prime factorization) – 2017-02-06
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    An abbreviation for id est, a Latin phrase meaning “that is.” It indicates that an explanation or paraphrase is about to follow: “Many workers expect to put in a forty-hour week — i.e., to work eight hours a day.” (Compare e.g.) – 2017-02-08
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If you look at integers mod 2, then we have $\{0,1\}$ and $0^2 =0$ and $1^2 = 1$. $0$ corresponds to the even numbers, $1$ to the odd ones.