# Modulo Arithmetic/Examples

## 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:

 $\displaystyle x$ $=$ $\displaystyle 0$ $\quad$ $\quad$ $\displaystyle x$ $=$ $\displaystyle 1$ $\quad$ $\quad$ $\displaystyle x$ $=$ $\displaystyle 3$ $\quad$ $\quad$ $\displaystyle x$ $=$ $\displaystyle 4$ $\quad$ $\quad$

### 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:

$a^2 + \paren {a + 2}^2 + \paren {a + 4}^2 + 1 \equiv 0 \pmod {12}$