17629 ÷ 9 = 1958 R7
Alright, let's work through the division of 17629 by 9 step by step.

Step 1:
9 goes into 1 0 times with a remainder of 1.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 1 to get 1.
Bring next digit (7) of the dividend behind the 1 and repeat the process: 17 / 9

Step 2:
9 goes into 17 1 times with a remainder of 8.
Write down 1 as next digit of of the result. 
Result so far: 1
Subtract 9 from 17 to get 8.
Bring next digit (6) of the dividend behind the 8 and repeat the process: 86 / 9

Step 3:
9 goes into 86 9 times with a remainder of 5.
Write down 9 as next digit of of the result. 
Result so far: 19
Subtract 81 from 86 to get 5.
Bring next digit (2) of the dividend behind the 5 and repeat the process: 52 / 9

Step 4:
9 goes into 52 5 times with a remainder of 7.
Write down 5 as next digit of of the result. 
Result so far: 195
Subtract 45 from 52 to get 7.
Bring next digit (9) of the dividend behind the 7 and repeat the process: 79 / 9

Step 5:
9 goes into 79 8 times with a remainder of 7.
Write down 8 as next digit of of the result. 
Result so far: 1958
Subtract 72 from 79 to get 7.

The final result is 1958 with a remainder of 7.