A RATHER STRANGE PROPERTY ABOUT NUMBERS

I saw this property in a magazine from 1978, in an issue by Alexandru LUPAȘ.

It is presented in the solution to Problem 16 667, on page 47.

I've extracted the essentials in the picture below.

I will write the solution on my Blog  : https://ogeometrie-cip.blogspot.com/ 

We have discussed case n=4 in detail. The general one mimics this one.