We make a long list of numbers in the following way. We start with \(1\) and \(1\). After that, each new number is the last digit of the sum of the two numbers right before it. For example, the beginning of the list is \[1,\,1,\,2,\,3,\,5,\,8,\,3,\,1,\,4,\ldots\]
Show that, if we keep making numbers like this forever, the list must eventually start repeating in a loop.