The link to the proof is https://www.simonellismaths.com/blog/new-maths (the link in he article says "post" instead of "blog"). It's standard modulo arithmetic, but it's nice to see a rediscovery by a kid.
Fun story showing how it's more fun to find things in math than just be told them. I think if more people had this kind of experience there would be a lot less hate on math! I have to say I like the subtraction method mentioned more though, both sides of the number end up staying smaller making it quicker to converge and calculate.