Is this procedure for $5^{300} \bmod 11$ correct?

Question

I'm new to modular exponentiation. Is this procecdure correct?

$$5^{300} \bmod 11$$

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

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$$5^{300} = 3 + 3 + 3 + 3 +3 +3 + 3 + 3 +3 +4 + 9$$

Answer 1

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.

Answer 2

You could also notice $11$ is prime, so by Fermat's Little Theorem $ 5^{10}\equiv 1 \bmod 11$

from here we get $5^{300}=(5^{10})^{30}\equiv1^{30}\equiv 1 \bmod 11$

Answer 3

I see what you were trying to do with adding the numbers at the end.

Multiply them instead, and take that product $\mod 11$.