## Thursday, October 07, 2021

### 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