We write down all the numbers from 1 to one million successively. Then we replace each number with the sum of its digits. Then we repeat this process until only single-digit numbers remain. Which digit appears most frequently?
Among the integers from 1 to 999,999, there are exactly 111,111 that are congruent to 1 modulo 9, 111,111 to 2, ..., 111,111 to 9. Since an integer is congruent to the sum of its digits modulo 9, we conclude that the digit 1 will be written 111,111 times, the digit 2 will be written 111,111 times, ..., and the digit 9 will be written 111,111 times. The number 1,000,000 will be written at the end as a 1: there will then be 111,112 times the digit 1 written, making it the most frequent digit.

## Remark 2.

Invariance is not limited to divisibility. We can easily imagine that a sum or a difference might be preserved, or even that a point remains unchanged through various geometric transformations!