Modulo Arithmetic/Examples/11 Divides 3^3n+1 + 2^2n+3
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Example of Modulo Arithmetic
- $11$ is a divisor of $3^{3 n + 1} + 2^{2 n + 3}$.
Proof
We have:
\(\ds 3^{3 n + 1}\) | \(=\) | \(\ds 3 \times 3^{3 n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times \paren {3^3}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 27^n\) |
Now:
\(\ds 27\) | \(\equiv\) | \(\ds 5\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 27^n\) | \(\equiv\) | \(\ds 5^n\) | \(\ds \pmod {11}\) | Congruence of Powers | |||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 \times 27^n\) | \(\equiv\) | \(\ds 3 \times 5^n\) | \(\ds \pmod {11}\) | Congruence of Product | |||||||||
\(\ds \leadsto \ \ \) | \(\ds 3^{3 n + 1}\) | \(\equiv\) | \(\ds 3 \times 5^n\) | \(\ds \pmod {11}\) |
Then we have:
\(\ds 2^{4 n + 3}\) | \(=\) | \(\ds 2^3 \times \paren {2^4}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8 \times 16^n\) |
and:
\(\ds 16\) | \(\equiv\) | \(\ds 5\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 16^n\) | \(\equiv\) | \(\ds 5^n\) | \(\ds \pmod {11}\) | Congruence of Powers | |||||||||
\(\ds \leadsto \ \ \) | \(\ds 8 \times 16^n\) | \(\equiv\) | \(\ds 8 \times 5^n\) | \(\ds \pmod {11}\) | Congruence of Product | |||||||||
\(\ds \leadsto \ \ \) | \(\ds 2^{4 n + 3}\) | \(\equiv\) | \(\ds 8 \times 5^n\) | \(\ds \pmod {11}\) |
So:
\(\ds 3^{3 n + 1} + 2^{4 n + 3}\) | \(\equiv\) | \(\ds \paren {3 \cdot 5^n + 8 \cdot 5^n}\) | \(\ds \pmod {11}\) | Modulo Addition is Well-Defined | ||||||||||
\(\ds \) | \(\equiv\) | \(\ds \paren {3 + 8} 5^n\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds \) | \(\equiv\) | \(\ds 11 \times 5^n\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds \) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod {11}\) |
Hence:
- $\forall n \in \N: 11 \divides \paren {3^{3 n + 1} + 2^{4 n + 3} }$
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Some Properties of $\Z$: Exercise $2.10$