Quotient: 9756097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097561

Remainder:  0

Start with the remainder.

number of zeros; Number;  the Number mod 41:

0 zeros: 41      0

1 zero:  401     32

2 zeros: 4001    24

3 zeros: 40001   26

4 zeros: 400001   5

5 zeros: 4000001   0

6 zeros: 40000001  32

There is a repeating pattern such that when the number of zeros is 100, the remainder is in fact zero.

The quotients make a series that starts:

1, 9, 97, 975, 9756, 97561, 975609, ...

Except for the first '1' corresponding to 0 zeros, the number of digits in the quotient equals the number of zeros in the "400..001 number".

Look at 4/41 as per https://www.mathsisfun.com/calculator-precision.html

0.097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097561

Now, erase the "0.0" and just look at the first 102 characters starting with the "9":

975609756097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097

Characters 100, 101, and 102 are  "097".

The 100-th digit is 0 but the next digit suggests converting this to "1" due to rounding.  

So the quotient should be:

9756097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097561

And this is confirmed at the mathsisfun website when the quotient is multiplied by 41.