Dividing Whole Numbers with a Remainder

A farmer has 125 eggs and wants to pack them in boxes of 6. How many boxes does he need? The quotient and the remainder of the division of 125 by 6 allow us to find the result: He needs 21 boxes. Twenty of them will be full and the 21st will contain just 5 eggs.
How do we carry out this calculation?

I. Definitions

Consider two positive whole numbers a and b, where b is not 0. When we divide a by b with a remainder we associate two other whole numbers q and r with them, such that a = bq + r, where r < b. In practice, we carry out the division of a by b with a remainder as follows:

a is called the dividend, b is the divisor, q is the quotient, and r is the remainder.

If we go back to the problem in the introduction, the division with remainder can be set out like this:

We have 125 = 6 × 20 + 5, where 5 < 6. So we can fill 20 boxes with 6 eggs and there are 5 eggs remaining, which need a 21st box.

II. Technique for dividing with a remainder

We can start with an example to describe the stages of the calculation: We want to find the quotient and the remainder when we divide 78 by 4.

How many times does 4 go into 7? 1 time.
1 × 4 = 4; 7 – 4 = 3
We move the 8 down.

How many times does 4 go into 38? 9 times.
9 × 4 = 36; 38 – 36 = 2

We don’t have any more digits to bring down, so at this stage the calculation stops.

We can write 78 = 4 × 19 + 2. The dividend is 78, the divisor 4, the quotient 19, and the remainder 2.

Note: If the divisor is greater than or equal to 10 (it is written with at least two digits) the technique is the same.

III. Notes

When we divide with a remainder, the remainder is less than the divisor.

Thus, even if it is true that 78 = 4 × 18 + 6, the diagram below is not a correct division of 78 by 4 with a remainder (because 6 is greater than 4).

When we divide 78 by 4 with a remainder, the quotient is 19 and the remainder is 2; this means that there are 4 non-zero multiples of 19 less than or equal to 17, the last one being 78 – 2, which is 76.

These multiples are: 19, 38, 57, and 76.

This can be generalized to all divisions with a remainder, by replacing 78 and 4 by any other numbers (and 19 and 2 by the corresponding quotient and remainder).

If the remainder is equal to 0, then the dividend is a multiple of the divisor. Equally, if the dividend is a multiple of the divisor, then the remainder will be equal to 0.

Some calculators will work out the quotient and the remainder of a division of two numbers with a remainder.

Read more:
Reading and Writing Decimal Numbers
Multiplying or Dividing a Decimal Number by 10, 100, or 1,000
Multiplying Decimal Numbers
Dividing Decimal Numbers
Converting Decimals to Fractions
The Effect of Addition and Multiplication on the Order of Numbers
Describing Different Types of Numbers
Comparing and Ordering Decimal Numbers
Adding and Subtracting Decimal Numbers