Multiplicative Order
Definition
Given a positive integer \(n\) and an integer \(a\) coprime to \(n\), the multiplicative order of \(a\) modulo \(n\) is the smallest positive integer \(k\) such that
$$
\begin{align*}
a^k \equiv 1 (\mod n)
\end{align*}
$$
Example
Suppose \(n = 7\) and \(a = 3\), then
$$
\begin{align*}
3^1 &= 3 \equiv 3 \mod n \\
3^2 &= 6 \equiv 6 \mod n \\
3^3 &= 9 \equiv 2 \mod n \\
3^4 &= 81 \equiv 4 \mod n \\
3^5 &= 243 \equiv 5 \mod n \\
3^6 &= 729 \equiv 1 \mod n \\
\end{align*}
$$
So the multiplicative order of 3 modulo 7 is 6. Similarly for a = 4 and n = 7
$$
\begin{align*}
4^1 &= 4 \equiv 4 \mod n \\
4^2 &= 16 \equiv 2 \mod n \\
4^3 &= 64 \equiv 1 \mod n \\
\end{align*}
$$
So the multiplicative order of 4 modulo 7 is 3.