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What does $10^{a} \equiv 1 \pmod{p}$ mean? Especially when relating it to base 10 referring to $10^{a}$?

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One consequence is that $1/p$ has a repeating decimal representation of period a divisor of $a$. – lhf Nov 2 '12 at 0:50

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 $10^{p-1}\equiv 1 \pmod p$.

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Can it be related to partitioning a number to segments of length a, sum the segments, and check for divisibility by p? – Ameen Nov 2 '12 at 9:20

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