<s>[INST] Modulo Equations I am trying to solve a problem involving modulo arithmetic but I am not sure what method to use as I have never done this style of question before nor do I have any examples to work from.
The question is:

Solve for $x$ where $x^5 = 11 \mod (35)$

So I though Eculid's GCD algorithm would help but I just get circular equations and get nowhere.  
Thanks
 [/INST] Let $x\in\mathbb{Z}$ be such that $x^5\equiv 11\pmod{35}$.  Then, $x^{15}=\left(x^5\right)^3\equiv 11^3\equiv 1\pmod{35}$.  Since $\gcd(x,35)=1$, $x^{12}=x^{\text{lcm}(5-1,7-1)}=x^{\lambda(35)}\equiv 1\pmod{35}$, where $\lambda$ is the Carmichael function.  Thus, $x^3=x^{\gcd(15,12)}=1\pmod{35}$, whence $1\equiv x^6\equiv x^5\cdot x\equiv 11\pmod{35}$, so that $x\equiv 11^{-1}\equiv 16\pmod{35}$, which is obviously a solution to this congruence.
Alternatively, $x^5\equiv 11\pmod{35}$ iff $x^5\equiv 1\pmod{5}$ and $x^5\equiv 4\equiv 3^4\pmod{7}$, which is equivalent to $x\equiv 1\pmod{5}$ and $x\equiv 3^2\equiv 2\pmod{7}$.  Consequently, $x^5\equiv 11\pmod{35}$ if and only if $x\equiv 16\pmod{35}$.
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