Use of language "the exponent to which $10$ belongs $\pmod p$" In her paper Periodic Decimal Fractions, Bennet uses the phrase "the exponent to which $10$ belongs modulo $p$" to mean "the least positive integer $k$ such that $10^k\equiv 1\pmod p$."
After looking around, it seems that this language is somewhat commonplace, and it can be found on documents by Wolfram and other papers. As a native English speaker, I have never encountered this language and it took me a while to figure out the author's intent.


*

*Does anybody know where this language came from?

*How can we explain the use of "belongs to" here?

 A: It comes from Gauss's Disquisitiones Arithmeticae, and his development of primitive roots.
In the section on primitive roots he proves, amongst others, two results:


*

*If $a^k\equiv 1 \pmod p$, then $k|(p-1)$

*The number of elements that (Gauss's word) 'belong' to $k$ is $\phi(k)$, with $\phi(k)$ being the totient function.
For example, using $p=19$ and primitive root $3$ we get the following table.
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\
\hline
3&9&8&5&15&7&2&6&18&16&10&11&14&4&12&17&13&1\\
\hline
\end{array}
The divisors of $p-1=18$ are $1,2,3,6,9,18$, with totient values $1,1,2,2,6,6$.
The order of an element is the lowest value of $k$ such that $a^k\equiv 1\pmod p$, which gives the following table:
\begin{array}{|c|c|c|}
\hline
k&\phi(k)&elements\\
\hline
1&1&18\\\hline
2&1&9\\\hline
3&2&7,11\\\hline
6&2&8,12\\\hline
9&6&4,5,6,9,16,17\\\hline
18&6&2,3,10,13,14,15\\
\hline
\end{array}
At this point, Gauss says that, for example, '$16$ belongs to the exponent $9$, or '$8$ belongs to the exponent $6$'.
To clear up, $7$ belongs to exponent $3$ because $7^3\equiv 1\pmod{19}$. If we look up $7$ in the first table, we can see that $3^6\equiv 7\pmod{19}$, and then that $(3^6)^3=3^{18}\equiv 1\pmod{19}$.
