Math problem that is driving the Internet crazy

[polymath257]

(bold mine)

I think I've spotted part of your problem.

Boru

There is no number higher than 9. So, 8 + 8 = 16 or, 1 + 6 = 7--DUH! Just like 16 + 16 = 32. Not so fast as 3 + 2 = 5 or, 7 + 7 = 14, 1 + 4 = 5 as there is no higher number than 9. Then 32 + 32 = 64. There is no number higher than nine so 6 + 4 = 10. There is no number higher than nine so, 1 + 0 = 1 and 64 + 64 = 128, 1 + 2 + 8 = 11 there is no higher number than nine so, 1 + 1 = 2. Get it? It's a fun toy to play with and it is infinite. 1,2,4,8,7,5,1,2,4,8,7,5 it never ends.

It's called modular arithmetic base 9. This works because the remainder when you divide 10 (the base of our number system) by 9, is 1.

There are a lot of 'magic tricks' that are based on this.

You can do similar thing with multiplication by 5 to get 5, 7, 8, 4, 2, 1, 5, 7, 8, 4, 2, 1,... Notice that these are your sequence with order reversed.

Or multiply by 7 each time: 7, 2, 5, 8, 2, 5, 8, ... Note that these are every other one from the previous.

Or you can take any two numbers, say 384 and 927. Multiply them to get 355968. Now, "reduce all by 9's": 384=15=6 and 927=18=9. Now, 6*9=54=9.
But, 355968=3+5+5+9+6+8=36->3+6=9, the same answer. And you will always get the same answer (but not always 9).