Proof: Characterize m

Question

Characterize $m$, an integer, such that $m^2≡1 \pmod{5}$. State your characterization as an "if and only if" statement and then prove it.

This question is on my study guide for a test that is on Friday (12/5). We are talking about Proof by Mathematical Induction, Strong PMI, and review of basics that we have covered so far.

I'm not sure how to start this proof.

Hint: you only have to look at numbers $\mod 5$, and you have finitely many cases to check. Do them one-by-one. — Wojowu, Dec 02 '14 at 18:16

score 1 · Answer 1 · answered Dec 02 '14 at 21:06

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A hint:

Any integer $m$ can be written in the form $m=5k+\ell$ with $k\in{\mathbb Z}$ and $\ell\in\{0,1,2,3,4\}$. Now square $m$ and check what the condition $m^2\equiv 1$ $({\rm mod}\>5)$ implies for $\ell$.

answered Dec 02 '14 at 21:06

Christian Blatter

226,825

Would it be sufficient to say that the statement holds, "iff m == a, b" and then just show ==> by inspection of a, b (which we know, but I'm being coy about), and then showing <== by inspection of the other three elements? – The Chaz 2.0 Dec 03 '14 at 16:16
(I mean stylistically - is it appropriate to demonstrate that it holds in two cases, and that it doesn't hold in the other three?) – The Chaz 2.0 Dec 03 '14 at 16:17
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@TheChaz2.0: After checking the five cases for $\ell$ you can safely say that exactly the $m$ of the form $m= 5k\pm1$ with $k\in{\mathbb Z}$ satisfy the condition $m^2\equiv1$ $>({\rm mod}\ 5)$. – Christian Blatter Dec 03 '14 at 18:44

Proof: Characterize m

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