# Modular Arithmetic with Negative Exponents

In one proof I'm trying to figure out, I've arrived at a part where I am thinking about whether $$x≡1$$ and $$y≡1$$ imply $$x^ay^b≡1$$ for any real integers a and b. I know that in general, if $$u≡v$$ and $$c≡d$$, then $$ub≡vd$$. But what if one of $$a$$ and $$b$$ is negative? Let's say that $$a≡1$$. Can you prove that $$a^k≡1$$ for any integer k, even if k could be negative?

• In modular arithmetic, $x^{-1} \pmod{n}$ means the residue $y\pmod{n}$ so that $xy \equiv 1\pmod{n}$, if it exists. – Jair Taylor Mar 12 at 6:50
• In particular, if one wants to prove that $a^{-j}\equiv1\pmod m$, one first needs to know the definition of the notation $a^{-j}\pmod m$. The proof must start from that! – Greg Martin Mar 12 at 7:18
• If $\,x^n\equiv 1$ for an integer $n\ge 1$ then $\,xx^{n-1}\equiv 1\,$ so $\,x^{-1}\equiv x^{n-1}\,$ so $\,x^{-k}\equiv x^{(n-1)k}\$ – Bill Dubuque Mar 14 at 3:10