95637 ÷ 6 = 15939 R3
Alright, let's work through the division of 95637 by 6 step by step.

Step 1:
6 goes into 9 1 times with a remainder of 3.
Write down 1 as next digit of of the result. 
Result so far: 1
Subtract 6 from 9 to get 3.
Bring next digit (5) of the dividend behind the 3 and repeat the process: 35 / 6

Step 2:
6 goes into 35 5 times with a remainder of 5.
Write down 5 as next digit of of the result. 
Result so far: 15
Subtract 30 from 35 to get 5.
Bring next digit (6) of the dividend behind the 5 and repeat the process: 56 / 6

Step 3:
6 goes into 56 9 times with a remainder of 2.
Write down 9 as next digit of of the result. 
Result so far: 159
Subtract 54 from 56 to get 2.
Bring next digit (3) of the dividend behind the 2 and repeat the process: 23 / 6

Step 4:
6 goes into 23 3 times with a remainder of 5.
Write down 3 as next digit of of the result. 
Result so far: 1593
Subtract 18 from 23 to get 5.
Bring next digit (7) of the dividend behind the 5 and repeat the process: 57 / 6

Step 5:
6 goes into 57 9 times with a remainder of 3.
Write down 9 as next digit of of the result. 
Result so far: 15939
Subtract 54 from 57 to get 3.

The final result is 15939 with a remainder of 3.