<s>[INST] Is this procedure for $5^{300} \bmod 11$ correct? I'm new to modular exponentiation. Is this procecdure correct?
$$5^{300} \bmod 11$$

$$5^{1} \bmod 11 = 5\\
5^{2} \bmod 11 = 3\\
5^{4} \bmod 11 = 3^2 \bmod 11 = 9\\
5^{8} \bmod 11 = 9^2\bmod 11 = 4\\
5^{16} \bmod 11 = 4^2 \bmod 11 = 5\\
5^{32} \bmod 11 = 5^2 \bmod 11 = 3$$

$$5^{300} = 3 + 3 + 3 + 3 +3 +3 + 3 + 3 +3 +4 + 9$$
 [/INST] You're almost there.  Instead of your last line, you want:
$$5^{300} \equiv 5^{4}5^{8}5^{32}5^{256} \pmod{11}$$
Now you replace each of those factors with the modular equivalent you found before (e.g. $5^4\implies 9$)
Let me know if you need more help.
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