Simultaneous congruence relations $x^a \equiv 1\pmod p$ and $x^b \equiv 1\pmod p$

Question

Let $x$ be a natural number and $p$ a positive prime such that $\gcd(x, p)=1$. If $x^a \equiv 1\pmod p$ and $x^b \equiv 1\pmod p$, can we derive a relation between $a$ and $b$?

Answer 1

The gcd condition that MXYMXY stated is the most general possible.

Your equations imply that that $x^{b\pm a} \equiv 1 \pmod p$, which happens if and only if $\mathrm{ord}(x)\mid b\pm a$. This, of course is equivalent to $\mathrm{ord}(x) \mid a$ and $\mathrm{ord}(x)\mid b$, which is in turn equivalent to $\mathrm{ord}(x)\mid \gcd(a,b)$.