111111111 x 111111111 = 12345678987654321

12 + 3 - 4 + 5 + 67 + 8 + 9 = 100

1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321

The number 142857 is rather special. When you multiply it the digits found in the answer stay the same, just in a different order. Check it out:

1 x 142857 = 142857
2 x 142857 = 285714
3 x 142857 = 428571
4 x 142857 = 571428
5 x 142857 = 714285
6 x 142857 = 857142

What happens when you multiply it by 7? You get another rather special number:

7 x 142857 = 999999

And it doesn't stop there:

142 + 857 = 999
14 + 28 + 57 = 99

142857 x 142857 = 20408122449
20408 + 122449 = 142857