Below is a listing of the quotient and remainder for the first several values for N where N is a series of 7s.

If k is the number of 7s, the remainder is 0 when k is 0 mod 8 and 7 when k is 1 mod 8.

Since 100001 is 1 mod 8, the remainder is 7.

The quotient, starting with N=5, is N-4 digits long with a repeating pattern of 4 7s followed by 4 0s, so a cycle of 8.  We already know 10001 is 1 mod 8, so 10001-4 is 5 mod 8

The quotient is [77770000] repeated 1249 times followed by [77770]

777700007777000077770000 ... repeated 1249 times in total ... 7777000077770

1 [0, 7]

2 [0, 77]

3 [0, 777]

4 [0, 7777]

5 [7, 7770]

6 [77, 7700]

7 [777, 7000]

8 [7777, 0]

9 [77770, 7]

10 [777700, 77]

11 [7777000, 777]

12 [77770000, 7777]

13 [777700007, 7770]

14 [7777000077, 7700]

15 [77770000777, 7000]

16 [777700007777, 0]

17 [7777000077770, 7]

18 [77770000777700, 77]

19 [777700007777000, 777]

20 [7777000077770000, 7777]

21 [77770000777700007, 7770]

22 [777700007777000077, 7700]

23 [7777000077770000777, 7000]

24 [77770000777700007777, 0]

25 [777700007777000077770, 7]

26 [7777000077770000777700, 77]

27 [77770000777700007777000, 777]

28 [777700007777000077770000, 7777]

29 [7777000077770000777700007, 7770]