Odd Power of 2 is Congruent to 2 Modulo 3

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Theorem

Let $n \in \Z_{\ge 0}$ be an odd positive integer.

Then:

$2^n \equiv 2 \pmod 3$


Proof

From Congruence Modulo 3 of Power of 2:

$2^n \equiv \paren {-1}^n \pmod 3$

We have that $n$ is odd.

Hence:

\(\displaystyle 2^n\) \(\equiv\) \(\displaystyle -1\) \(\displaystyle \pmod 3\)
\(\displaystyle \) \(\equiv\) \(\displaystyle 3 - 1\) \(\displaystyle \pmod 3\)
\(\displaystyle \) \(\equiv\) \(\displaystyle 2\) \(\displaystyle \pmod 3\)

$\blacksquare$