I don't know if it qualifies as a "mathematical coincidence" or not, but I find it amusing to demonstrate to kids that you can multiply 142,857 by 1, 2, 3, etc., and the result is always a number with the same digits but in a different order:
2 × 142,857 = 285,714
3 × 142,587 = 428,571
4 × 142,587 = 571,428
Etc. But then, suddenly:
7 × 142,857 = 999,999
