Definition
A Full Reptend Prime is a prime \(p\) for which \(1/p\) has \(p-1\) digits in its decimal expansion. Moreover, a prime \(p\) is full reptend if and only if \(10\) is a primitive root modulo \(p\). This means that
$$
\begin{align*}
10^k \equiv 1 \pmod{p}
\end{align*}
$$
for \(k = p - 1\) and not for any smaller than \(k\). In other words, the multiplicative order of \(10\) modulo \(p\) is \(p-1\).
Example
Recall that the multiplicative order of \(a\) when \(a\) is coprime to \(n\), is the smallest \(k\) such that \(a^k \equiv 1 \pmod{n}\). Here set \(n = p\) and \(a = 10\), the multiplicative order of \(10\) modulo \(p\) is then the smallest \(k\) for which \(10^k \equiv 1 \mod p\).
As an example, set \(p = 7\), observe that
$$
\begin{align*}
10^1 &\equiv 3 \mod 7 \\
10^2 &\equiv 2 \mod 7 \\
10^3 &\equiv 6 \mod 7 \\
10^4 &\equiv 4 \mod 7 \\
10^5 &\equiv 5 \mod 7 \\
10^6 &\equiv 1 \mod 7 \\
\end{align*}
$$
So the multiplicative order of \(10\) modulo \(7\) is \(6\) which is \(7-1\). So \(7\) is a full reptend prime.