[PretzelPirate]

> naturally suggests multiplying by 11

Is this a named concept that I can learn about?

[zahlman]

The repeating pattern has 2 digits, and is a multiple of 9. 9 times 11 is 99, so that multiplication gives a "repeating pattern of a multiple of 99" — equivalently, a multiple of a repeating pattern of 99. And I assume you're familiar with the concept of .99999... equaling 1.

[Handy-Man]

Not sure if this helps but check it out

https://mathcentral.uregina.ca/QQ/database/QQ.09.07/h/jack1....