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:
\(\ds 2^n\) | \(\equiv\) | \(\ds -1\) | \(\ds \pmod 3\) | |||||||||||
\(\ds \) | \(\equiv\) | \(\ds 3 - 1\) | \(\ds \pmod 3\) | |||||||||||
\(\ds \) | \(\equiv\) | \(\ds 2\) | \(\ds \pmod 3\) |
$\blacksquare$