What's so special about 6987952709035199279905674?

Éric Angelini came up with this yesterday:

abcdefghij... (letters are distinct digits):
a is divisible by 1 with remainder 0
ab is divisible by 2 with remainder 1
abc is divisible by 3 with remainder 2
abcd is divisible by 4 with remainder 3
etc.

The largest such integer would be 95674103:

9/1 leaves a remainder of 0

95/2 leaves a remainder of 1

956/3 leaves a remainder of 2

9567/4 leaves a remainder of 3

95674/5 leaves a remainder of 4

956741/6 leaves a remainder of 5

9567410/7 leaves a remainder of 6

95674103/8 leaves a remainder of 7

Much more interesting is if we dispense with the "distinct digits" limitation. Let us allow any digit. The largest integer is now 6987952709035199279905674:

6/1 leaves a remainder of 0

69/2 leaves a remainder of 1

698/3 leaves a remainder of 2

6987/4 leaves a remainder of 3

69879/5 leaves a remainder of 4

698795/6 leaves a remainder of 5

6987952/7 leaves a remainder of 6

69879527/8 leaves a remainder of 7

698795270/9 leaves a remainder of 8

6987952709/10 leaves a remainder of 9

69879527090/11 leaves a remainder of 10

698795270903/12 leaves a remainder of 11

6987952709035/13 leaves a remainder of 12

69879527090351/14 leaves a remainder of 13

698795270903519/15 leaves a remainder of 14

6987952709035199/16 leaves a remainder of 15

69879527090351992/17 leaves a remainder of 16

698795270903519927/18 leaves a remainder of 17

6987952709035199279/19 leaves a remainder of 18

69879527090351992799/20 leaves a remainder of 19

698795270903519927990/21 leaves a remainder of 20

6987952709035199279905/22 leaves a remainder of 21

69879527090351992799056/23 leaves a remainder of 22

698795270903519927990567/24 leaves a remainder of 23

6987952709035199279905674/25 leaves a remainder of 24