Arithmetic is the branch of mathematics that deals with the study of numbers using various mathematical operations like addition, subtraction, division, and multiplication and all kinds of real numbers, including integers. Four fundamental arithmetic operations form the foundation of mathematics. These are addition, subtraction, multiplication, and division. The meaning of “division” is to break or divide a number into equal parts. The four main terms of the division operation are divisor, dividend, quotient, and remainder. One can explain division as the process of repetitive subtraction. It is basically a mechanism for grouping things equally. For instance, dividing or grouping students into several rows in morning drills in army training schools.

Let’s take the example of dividing a number A by another number B. If A is divided by B, then A is the dividend and B is the divisor. Here we have two possibilities:

• Either the remainder is left, which is the division with the remainder, or
• No remainder is left, which is division without remainder (meaning B completely divides A or R = 0).

In this math article, we are going to discuss division with remainders along with a brief description of remainders.

What is Remainder?

Remainder, as the name itself suggests, is something that is left over or remaining. Actually, the word got its origin from the French word “remaindre’” which literally means to remain. Let’s recall that division involves four major terms: the divisor, the dividend, the quotient, and the remainder.

In the case when a number (called the dividend) is not completely divided by another number (which is called the divisor), at the end we are left with a value after the division process is completed. This value is referred to as the “remaining” or “remainder.” The remainder is basically the portion of the dividend that cannot be fairly divided by the divisor.

Now, coming to the division algorithm, we have,

Dividend = Divisor × Quotient + Remainder

We can obtain the remainder as,

Remainder = Dividend – (Divisor × Quotient)

provided that the divisor, dividend, and quotient are given.

Properties of Remainder

Here are some of the properties of the remainder:

• When a number is divided completely, the remainder is zero.
• When a number is divided by its factor, the remainder comes out to be zero.
• The remainder obtained at the end of the division must be less than the divisor. In the case where the remainder is more than the divisor, it indicates that there is still room for more division.
• Reminder is always a non-negative number, which means it can be 0 or any positive number but never a negative number.

What is division with remainders?

We will try to figure out this concept using examples. Before moving directly to division with remainder, let’s first understand what actually is division without remainder.

Let’s consider that you have 10 dollars to distribute equally among 5 children so that there is no conflict among them. Upon dividing 10 (the amount) by 5 (the number of children), you will find that each child will get 2 dollars, and at the end, you are left with zero dollars. This is what we refer to as “division without remainder.”

Now coming to the second scenario After some time, two more children join the group; now there are seven in total. But you still have 10 dollars to distribute. So you again followed the same method, but this time the divisor is 7 in place of 5. So upon dividing 10 by 7, you find that each child will get 1 dollar, and at the end, you are left with 3 dollars. These 3 dollars are not enough to be distributed equally among 7 kids and are thus referred to as the “remainder.” This is what we refer to as “division with remainder.”

Let’s investigate what happened in these two cases. You will find that 5 is the factor of 10, so it completely divides it without leaving any remainder. On the other hand, 7 is not a factor of 10, thus leading to some remainder at the end.

Hence, you can easily conclude that dividing a number by its non-factor number will always lead to a division with a remainder.

How to work out division with remainders

Finding the remainder upon division is a very easy task. You just need to keep dividing the dividend by some multiple of the divisor. But the question is, until when do we need to repeat this procedure? The procedure needs to be repeated until the number obtained at the end is less than the divisor. Let’s have a look at an example here,

Here in the above example, you will observe that the value obtained at the end is 3, which is less than the divisor 4. Thus, this marks the end of the division process, and 3 is the remainder.

Summary

Let’s summarize whatever we have discussed till now. It will help you in quickly revising the important concepts once you have gone through the complete content.

• Arithmetic is the branch of mathematics that deals with the study of numbers using various mathematical operations like addition, subtraction, division, and multiplication of all kinds of real numbers, including integers.
• The remainder is basically the portion of the dividend that cannot be fairly divided by the divisor.
• Division can be checked using division algorithm which can be expressed as,

Dividend = Divisor × Quotient + Remainder

Solved Questions

Example 1. When 20 is divided by 10, check whether it is division with remainder or the division without remainder?

Solution 1. Clearly, we found that 10 is the factor of 20. Thus upon dividing 20 by 10 we are left with zero as remainder. Hence it is division without remainder.

Example 2. When 35 is divided by 6, check whether it is division with remainder or the division without remainder?

Solution 2. Upon division of 35 by 6, we obtained,

35 = 6 × 5 + 5

Here, 5 is the remainder. Also 6 is not a factor of 35. Hence it is a division with the remainder.

Example 3. Determine remainder when we have quotient = 8, divisor =15 and divided= 125.

Solution 3. Clearly we know that,

Remainder = Dividend – Divisor × Quotient

on plugging the values in the above equation, we obtained,

R = 125 – 8 × 15 = 5

Hence, we obtained 5 as the remainder in this case.

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