Modulo Arithmetic/Examples/Residue of 2^512 Modulo 5
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Example of Modulo Arithmetic
The least positive residue of $2^{512} \pmod 5$ is $1$.
Proof
\(\ds 2^4\) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod 5\) | as $2^4 = 16 = 3 \times 5 + 1$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {2^4}^{128}\) | \(\equiv\) | \(\ds 1^{128}\) | \(\ds \pmod 5\) | Congruence of Powers | |||||||||
\(\ds \leadsto \ \ \) | \(\ds 2^{512}\) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod 5\) |
Hence the result.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $2$. Equivalence Relations: Exercise $6$