Division by complements is very easy. First let us see how division by 9 can be done. For 9 the complement is 1, so it is the easiest. Let us start with an example: 43 / 9
Write the divisor 9 on the left and the dividend 43 on the right of the line. Write the complement of 9 below it in the divisor part. Since the base is 10, which has one zero, we set aside one digit at the end for the remainder
9 4 3
1
Write the left most digit of the dividend (4) below the line
9 4 3
1
4
Multiply that by the complement (4 x 1) and write the answer below the next digit (3) of the dividend
9 4 3
1 4
4
Add the last line of digits (3 + 4) and write the answer (7) below the line
9 4 3
1 4
4 7
Answer: 4 is the quotient and 7 is the remainder
Division by 9 is very simple but to get a clear idea of the method we need to try other numbers. Let us divide a number by 8.
103 / 8
Write the divisor 8 outside and the dividend 43 inside. Write the complement of 8 below it in the divisor part. Set aside the last digit for the remainder
Write the left most digit of the dividend (1) below the line
8 1 0 3
2
1
Multiply 1 by the complement (2) and write the answer below the next digit of the dividend (0)
8 1 0 3
2 2
1 2
Add 0 + 2 and write the answer below. Multiply that by the complement (2) and write it below the next digit (3)
8 1 0 3
2 2 4
1 2
Add the digits in the last column (3 + 4) and write it below
8 1 0 3
2 2 4
1 2 7
Answer: 12 is the quotient and 7 is the remainder
In these two examples there were no carry overs and the sum of the numbers in the last column was less than the divisor, that made the calculation direct. Now let us look at another example that is not so direct.
233 /8
Set aside the last digit for the remainder and write the complement (2) of the divisor (8) below it
8 2 3 3
2
Write the first digit below the line
8 2 3 3
2
2
Multiply it by the complement and write it below the next digit
8 2 3 3
2 4
2
Next add 3 and 4
8 2 3 3
2 4
2 7
Now multiply 7 by the complement 2 and write it below the next digit (3)
8 2 3 3
2 4 14
2 7
Add 3 and 14 and write down the answer
8 2 3 3
2 4 14
2 7 17
We see that the remainder 17 is greater than 8. So reduce it by 8. 17 = 8 x 2 + 1
Add the number of 8s that occur in 17, (2) to the quotient. 17 + 2 = 19
That leaves 1 in the remainder column
Answer: 19 is the quotient and 1 is the remainder
Now you can try these,
(1) 123 /9, (2) 245 / 9, (3) 211 / 8
(4) 122/ 8 5) 236 / 8
This method is useful when we do division by numbers close to a base like 9, 8, 7; or 99, 98, 97 where the numbers are large but the complements are small.