## Divide by 1-Digit Divisors

The standard algorithm for long division is also based on our base-ten positional numeration system. As a first step toward understanding the algorithm for dividing D by d—that is, for solving , students must consider the process of distributing a set of D objects into d smaller sets so that each of the d sets contains Q objects and a remainder of R objects is left over. If this is possible, then

D = d × Q + R.

Here D is called the dividend, d is called the divisor, and the process of dividing D by d leads to a quotient Q and a remainder R. For Q to be as large as possible, 0 ≤ R < d. With this condition, there is just one answer to the division problem.

Example:

Distribute 17 objects equally among 5 smaller sets (D = 17 and d = 5). Try putting 2 objects in each of the 5 smaller sets.

In this case Q = 2 and R = 7, corresponding to17 = 5 × 2 + 7.

Because R > d (that is, 7 > 5), the quotient Q = 2 is not as great as possible. By distributing the 17 objects among 5 smaller sets with 3 objects in each set, the result becomes 17 = 5 × 3 + 2.

Now R < d (that is, 2 < 5) and no greater value of Q can be used.

To develop a systematic procedure for division that takes advantage of the base-ten system, the preceding process is applied to a number in expanded form, starting at the greatest place and working through the ones place.

For example, divide 597 by 4.

597 ÷ 4 or

First divide the 500 by 4.

5 hundreds = (4 × 1 hundred) + 1 hundred.

The remaining 1 hundred is regrouped as 10 tens and added to the 9 tens in the dividend to give a total of 19 tens. Dividing 19 tens, or 190, by 4, the result is

19 tens = (4 × 4 tens) + 3 tens.

The remaining 3 tens is regrouped as 30 ones and added to the 7 ones in the dividend to give a total of 37 ones. Dividing 37 by 4, the result is

37 = 4 × 9 + 1.

Combining these steps gives

597 = 500 + 90 + 7
= (4 × 100 + 100) + 90 + 7
= 4 × 100 + 190 + 7
= 4 × 100 + (4 × 40 + 30) + 7
= 4 × 100 + 4 × 40 + 37
= 4 × 100 + 4 × 40 + (4 × 9 + 1)
= (4 × 100 + 4 × 40 + 4 × 9) + 1
= 4 × (100 + 40 + 9) + 1
= 4 × 149 + 1

It is this sequence of steps that is carried out in implementing the long division algorithm

to obtain Q = 149 and R = 1. The fact that 597 = 4 × 149 + 1

provides a check of the correctness of our answer: 597 ÷ 4 = 149 R1.

Teaching Model 22.2: Model Division With Remainders