We’ll begin by refreshing the following conditions that are respected.

In remainder problems, when an integer n is divided by a nonzero integer d:

• n must be an integer,
• d must be a non-zero integer, d ? 0
• the quotient must be an integer,
• the remainder must be a non-negative integer, i.e. remainder ? 0

When you divide 19 (n) by 5 (d), you get a quotient of 3 and a remainder of 4.

The remainder, as you would know, is the amount left over after division.

The remainder is also defined as the distance from n to the nearest multiple of d that is smaller than the n.

Let’s take an example to understand this definition of the remainder.

19 cannot be divided exactly by 5.

In this case, the n is 19 and the d is 5.

To make it easier, let’s break this into 3 steps:

• Step 1: List the multiples of d (5) that are less than n (19). Those are …, -15, -10, -5, 0, 5, 10, 15, ….
• Step 2: Select the multiple that is nearest to n. In this case that is 15.
• Step 3: Find the distance from the n (19) to the multiple you selected in step 2 (15).

19 – 15 = 4. This is your remainder.

The figure below illustrates this.

You can also find the remainder by using the long division. This will save you some time on the question.

What is the remainder when dividing 5 by 8?

In this case, n is 5 and d is 8. (n < d)

Let’s again follow the 3 step process we described above.

• Step 1: List the multiples of d (8) that are less than the n (5). Those are …, -24, -16, -8, 0, ….
• Step 2: Select the multiple that is nearest to n. In this case that is 0.
• Step 3: Find the distance from n (5) to the multiple you selected in step 2 (0).

5 – 0 = 5. This is your remainder.

Again, you can also use the long division to find the remainder.

A couple of observations here.

If n is less than d then,

• quotient = 0
• remainder = n

You might do the long division as follows:

While this is the correct way to do the long division, this results in a decimal value for the quotient. Since the quotient must always be an integer, therefore this is not the correct way to find the remainder.

To find the remainder, the quotient must be an integer. In this case, the only possible integer value of the quotient is zero.

If you like analogies, consider the same problem in a real-life context;

“You have 5 oranges, there are 8 people, and each person should get the same number of oranges (This is the same as saying 5 ÷ 8, i.e. divide 5 oranges amongst 8 people). What is the number of oranges that each person gets (quotient) and how many oranges would remain undistributed (remainder)?”.

The obvious answer is that no one would get an orange, i.e. quotient =0

If no one gets any oranges, then you’ll be still left with all the 5 oranges, i.e. remainder = 5.

To summarise:

If n < d then,

• quotient = 0
• remainder = n
• You should like analogies 🙂