# Giuga Number/Examples/858

## Example of Giuga Number

$858$ is a Giuga number:

$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 {11} + \dfrac 1 {13} - \dfrac 1 {858} = 1$

## Proof

We have that:

$858 = 2 \times 3 \times 11 \times 13$

Then:

 $\displaystyle \dfrac 1 2 + \dfrac 1 3 + \dfrac 1 {11} + \dfrac 1 {13}$ $=$ $\displaystyle \frac {3 \times 11 \times 13 + 2 \times 11 \times 13 + 2 \times 3 \times 13 + 2 \times 3 \times 11} {2 \times 3 \times 11 \times 13}$ $\displaystyle$ $=$ $\displaystyle \frac {429 + 286 + 78 + 66} {858}$ $\displaystyle$ $=$ $\displaystyle \frac {859} {858}$ $\displaystyle$ $=$ $\displaystyle 1 + \frac 1 {858}$

Hence the result by definition of Giuga number.

$\blacksquare$