\(\text{结论:}\)
\[\exists x\in N^{*},a^x=1(\bmod m) \iff \gcd(a,m)=1
\]
\(\text{证明(右推左):}\)
\(\text{由欧拉定理可知当}\ x=\varphi(m)\ \text{时该同余方程成立。}\)
\(\text{证毕。}\)
\(\text{证明(左推右):}\)
\(\text{易知}\) \(\exists x\in N^{*},\gcd(a^x,m)=\gcd(a^x,a^x\mod m)=\gcd(a^x,1)=1\)
\(\text{假设}\) \(\gcd(a,m)\neq1\)，\(\text{则}\) \(\forall k\in N^{*}, \gcd(a^k,m)\neq1\)
\(\text{所以}\) \(\forall k\in N^{*}, \gcd(a^k,a^k\mod m)\neq 1,\) \(\text{与前者}\) \(\exists x\in N^{*},\gcd(a^x,m)=1\) \(\text{矛盾}\)
\(\text{所以必有}\) \(\gcd(a,m)=1\)
\(\text{证毕。}\)