Prove by either direct proof or contraposition I have a question like this: By direct proof or by contraposition: Let $a \in Z$, if $a \equiv 1 \pmod{5}$, then $a^2 \equiv 1 \pmod{5}$. Hypothesis: $a \in Z,~a \equiv 1 \pmod{5}$ Conclusion: $a^2 \equiv 1 \pmod{5}$ I am lost, I have no idea how to prove this. Tried contraposition, but this was as far as I got. Contraposition: If $a^2 \neq 1 \pmod{5}$, then $a \neq 1 \pmod{5}$

**Direct Proof:**

Using the property of Modular arithmetic,

If $x\equiv{c}\pmod{d}$, where $x,c,d\in{\Bbb{Z}}$,

then $x^n\equiv{c^n}\pmod{d}$, where $n\in{\Bbb{Z}}$

Since $a\equiv{1}\pmod{5}$

So $a^2\equiv{1^2}\equiv1\pmod{5}$

**Non-modular Proof**

Let $a=5k+1$, $k\in{\Bbb{N}}$

$a^2=(5k+1)^2=25k^2+10k+1=5(5k^2+2k)+1=5P+1$, $P\in{\Bbb{N}}$

So $a^2\equiv1\pmod{5}$