<s>[INST] Compute $5^{10,000}\equiv \mod 52$ I'm unsure how to solve questions like these, we are given an example below but I'm still confused about how they got from $5^{100\times10+0}$ to $5^{0}$.
ie) Compute $5^{1000} \bmod 77$
$$
5^{1000}=5^{(166⋅6+4)}=5^4=25^2=4^2=16=2 \pmod 7\\
5^{1000}=5^{(100⋅10+0)}=5^0=1 \pmod{11}\\
5^{1000}=2⋅2⋅11+1⋅8⋅7=44+56=100=23 \pmod{77}
$$
 [/INST] What you have to understand from the examples is that if $n$ is relatively prime to $m$, then there exists a exponent $k$ such that $n^k\equiv1\mod m$.
One exponent that works every time is $m-1$, but there may be others (divisors of $m-1$). For example, $5^6\equiv1\mod 7$, so 
$$5{(166.6+4}\equiv {(5^6)}^{166}.5^4\equiv 1^{166}.5^4\equiv5^4\mod7$$
So now you have to find the first power of $5$ equivalent to $1$ modulo $52$, and use the same principle.
Hint : you should find $5^1000\equiv1\mod52$ ;-)
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