Q: Modular equivalence for base ten What does $10^{a} \equiv 1 \pmod{p}$ mean? Especially when relating it to base 10 referring to $10^{a}$?

A: It means that if you divide $10^a$ by $p$, the remainder is $1$.
More generally, $A\equiv B\pmod m$ iff $m|A-B$.
Note that, modulo $p$ we have only $p$ remainders, and that keeping on multiplying by $10$ mod $p$ is cyclic. If $p\ne 2,5$ prime, then $10^{p-1}\equiv 1 \pmod p$.
