i want to prove $x^a \equiv x^{a\,\bmod\,8} \pmod{15}$.....(1) my logic:
here, since $\mathrm{gcd}(x,15)=1$, and $15$ has prime factors $3$ and $5$ (given) we can apply Euler's theorem.
we know that $a= rem + 8q$, where $8= \phi(15)$,
$x^a \equiv x^{rem}. (x^8)^q \pmod{15}$......(2)
applying Euler's theorem we get:
$x^a \equiv x^{rem} \pmod{15}$......(3)
Is this proof correct or should I end up in getting $x^a \equiv x^a \pmod {15}$...(4)