13663 ÷ 9 = 1518 R1
Alright, let's work through the division of 13663 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 (3) of the dividend behind the 1 and repeat the process: 13 / 9

Step 2:
9 goes into 13 1 times with a remainder of 4.
Write down 1 as next digit of of the result. 
Result so far: 1
Subtract 9 from 13 to get 4.
Bring next digit (6) of the dividend behind the 4 and repeat the process: 46 / 9

Step 3:
9 goes into 46 5 times with a remainder of 1.
Write down 5 as next digit of of the result. 
Result so far: 15
Subtract 45 from 46 to get 1.
Bring next digit (6) of the dividend behind the 1 and repeat the process: 16 / 9

Step 4:
9 goes into 16 1 times with a remainder of 7.
Write down 1 as next digit of of the result. 
Result so far: 151
Subtract 9 from 16 to get 7.
Bring next digit (3) of the dividend behind the 7 and repeat the process: 73 / 9

Step 5:
9 goes into 73 8 times with a remainder of 1.
Write down 8 as next digit of of the result. 
Result so far: 1518
Subtract 72 from 73 to get 1.

The final result is 1518 with a remainder of 1.