Giuga Number/Examples/858

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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.

$\blacksquare$


Sources