Exercise 2.1.13 (Reduced residue system modulo $7$ composed entirely of powers of $3$)

Question

Exhibit a reduced residue system modulo $7$ composed entirely of powers of $3$.

Richard Ganaye · Accepted Answer

\begin{proof} The following array gives $3^k \pmod 7$ for $1\leq k \leq 6$.
$$
\begin{array}{c|cccccc}
k & 1 & 2 & 3 & 4 & 5 & 6\
3^k \mod 7 & 3& 2& 6& 4& 5& 1
\end{array}
$$
Since $\{1,2,3,4,5,6\}$ is a reduced residue system modulo $7$,
$$\{3^1, 3^2 ,3^3 , 3^4,3^5,3^6\}$$
is a reduced residue system modulo $7$ composed entirely of powers of $3$.
\end{proof}

Exercise 2.1.13 (Reduced residue system modulo $7$ composed entirely of powers of $3$)

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Exercise 2.1.13 (Reduced residue system modulo $7$ composed entirely of powers of $3$)

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