35624 ÷ 7 = 5089 R1
Alright, let's work through the division of 35624 by 7 step by step.

Step 1:
7 goes into 3 0 times with a remainder of 3.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 3 to get 3.
Bring next digit (5) of the dividend behind the 3 and repeat the process: 35 / 7

Step 2:
7 goes into 35 5 times with a remainder of 0.
Write down 5 as next digit of of the result. 
Result so far: 5
Subtract 35 from 35 to get 0.
Bring next digit (6) of the dividend behind the 0 and repeat the process: 6 / 7

Step 3:
7 goes into 6 0 times with a remainder of 6.
Write down 0 as next digit of of the result. 
Result so far: 50
Subtract 0 from 6 to get 6.
Bring next digit (2) of the dividend behind the 6 and repeat the process: 62 / 7

Step 4:
7 goes into 62 8 times with a remainder of 6.
Write down 8 as next digit of of the result. 
Result so far: 508
Subtract 56 from 62 to get 6.
Bring next digit (4) of the dividend behind the 6 and repeat the process: 64 / 7

Step 5:
7 goes into 64 9 times with a remainder of 1.
Write down 9 as next digit of of the result. 
Result so far: 5089
Subtract 63 from 64 to get 1.

The final result is 5089 with a remainder of 1.