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To solve your example:

38  ) 4567

Work on the first two digits of 4567:

38  ) 4567
        45

Try the biggest multiple of 38 that fits into 45, which is 1. Write down the 1 above, and subtract the 38 (45-38=7):

          1
38  ) 4567
        45
       -38
          7

Now, bring down the next digit, 6:

          1
38  ) 4567
        45
       -38
          76

Now find the biggest multiple of 38 that fits into 76, which is 2 (and by luck is exaclty 76):

          12
38  ) 4567
        45
       -38
          76
          76
            0

Now, bring down the last digit, 7:

          12
38  ) 4567
        45
       -38
          76
          76
            07

This is the last digit, and 38 cannot fit into this, so it is the "remainder":

          120  Remainder 7
38  ) 4567
        45
       -38
          76
          76
            07

So, 4567/38 = 120 with a remaidner of 7/38

My calculator gives 4567/38 as 120.18, so it looks right.

Hows that?