5642 ÷ 9 = 626 R8
Alright, let's work through the division of 5642 by 9 step by step.

Step 1:
9 goes into 5 0 times with a remainder of 5.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 5 to get 5.
Bring next digit (6) of the dividend behind the 5 and repeat the process: 56 / 9

Step 2:
9 goes into 56 6 times with a remainder of 2.
Write down 6 as next digit of of the result. 
Result so far: 6
Subtract 54 from 56 to get 2.
Bring next digit (4) of the dividend behind the 2 and repeat the process: 24 / 9

Step 3:
9 goes into 24 2 times with a remainder of 6.
Write down 2 as next digit of of the result. 
Result so far: 62
Subtract 18 from 24 to get 6.
Bring next digit (2) of the dividend behind the 6 and repeat the process: 62 / 9

Step 4:
9 goes into 62 6 times with a remainder of 8.
Write down 6 as next digit of of the result. 
Result so far: 626
Subtract 54 from 62 to get 8.

The final result is 626 with a remainder of 8.