If $\displaystyle x^6 \equiv y^6 \mod{7^6}$, does this imply $\displaystyle x\equiv y \mod{7}$?
If $\displaystyle x^6 \equiv y^6 \mod{7^6}$, does this imply $\displaystyle x\equiv y \mod{7}$?
Of course not: $\displaystyle (-1)^6=1^6\!\!\!\pmod{7^6}\,,\,\,but\,\,\,-1\neq 1\!\!\!\pmod 7$