Modulo Arithmetic/Examples
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Examples of Modulo Arithmetic
$11$ divides $3^{3 n + 1} + 2^{2 n + 3}$
- $11$ is a divisor of $3^{3 n + 1} + 2^{2 n + 3}$.
Solutions to $x^2 = x \pmod 6$
The equation:
- $x^2 = x \pmod 6$
has solutions:
\(\ds x\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds x\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds x\) | \(=\) | \(\ds 3\) | ||||||||||||
\(\ds x\) | \(=\) | \(\ds 4\) |
Multiplicative Inverse of $41 \pmod {97}$
The inverse of $41$ under multiplication modulo $97$ is given by:
- ${\eqclass {41} {97} }^{-1} = 71$
Residue of $2^{512} \pmod 5$
The least positive residue of $2^{512} \pmod 5$ is $1$.
$n \paren {n^2 - 1} \paren {3 n - 2}$ Modulo $24$
- $n \paren {n^2 - 1} \paren {3 n + 2} \equiv 0 \pmod {24}$
$a^2 + \paren {a + 2}^2 + \paren {a + 4}^2 + 1$ Modulo $12$
Let $a$ be an odd integer.
Then:
- $a^2 + \paren {a + 2}^2 + \paren {a + 4}^2 + 1 \equiv 0 \pmod {12}$