Giuga Number/Examples/1722

Example of Giuga Number

$1722$ is a Giuga number:

$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 7 + \dfrac 1 {41} - \dfrac 1 {1722} = 1$

We have that:

$1722 = 2 \times 3 \times 7 \times 41$

Then:

$\ds \dfrac 1 2 + \dfrac 1 3 + \dfrac 1 7 + \dfrac 1 {41}$

$=$

$\ds \frac {3 \times 7 \times 41 + 2 \times 7 \times 41 + 2 \times 3 \times 41 + 2 \times 3 \times 7} {2 \times 3 \times 7 \times 41}$

$\ds $

$=$

$\ds \frac {861 + 574 + 246 + 42} {1722}$

$\ds $

$=$

$\ds \frac {1723} {1722}$

$\ds $

$=$

$\ds 1 + \frac 1 {1722}$

Hence the result by definition of Giuga number.

$\blacksquare$