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Mathematically this is a nothing, but it's fun and I hadn't seen it before. Pick a number, say 852143, and multiply it by 99, giving $$84362157.$$ Now break this up into twodigit chunks: $$84\qquad 36\qquad 21\qquad 57$$ (If there is an odd number of digits, put the leftmost one in a chunk by itself.) Add up the chunks: $$84 + 36 + 21 + 57 = 198$$ Repeat the process: $$1 + 98 = 99$$ You will always finish at 99. This same method can be used to test a number to see if it is divisible by 99: you can start with any number you like, and you will end at 99 if and only if the number you started with is a multiple of 99. (If the starting and ending numbers are !!s!! and !!e!!, then !!s\equiv e\pmod{99}!!.) The trick is, of course, completely analogous to the test for divisibility by 9. Maybe you know some mathematical kid who has recently learned the divisibilityby9 test and would enjoy seeing this version.
