Remainder

Introduction

Division is the basic arithmetical operation in mathematics. It is the process of distributing things into equal parts. When the dividend is not completely divisible by the divisor it left a number or the value is known as the remainder. In mathematics, in numerous instances, the leftover part or value is known as residues. Sometimes in many calculations residues are neglected or round off to obtain the answer in the whole number. For example, in the decimal number 5.04, the number after the decimal is 4. It represents remainder or residue. Sometimes to give only a whole number answer, the residue 4 is neglected to answer 5. When a polynomial is divided by a linear polynomial remainder theorem is used. In this tutorial, we are going to study Euclid’s division algorithm.

Division

Division is the basic arithmetical operation in mathematics. The division is the process of distributing things into equal parts. It is the inverse process of multiplication. The division can be represented by a symbol, / or sometimes it is denoted by ( ) for example $\mathrm{8\:\div\:2\:=\:4}$. This can be written as in multiplication form $\mathrm{2\:\times\:4\:=\:4}$. So we can say that it is the inverse operation of multiplication.

Basic terminologies regarding division:

Dividend

It is any number or amount or value that we divide or distribute into equal parts known as the dividend. It is an important part of the division.

Divisor

It is a number that divides the dividend called a divisor.

Quotient

It is a value or answer which is obtained from the performing division known as a quotient.

Remainder

After performing division, the value that remains or is left out is known as the remainder

For example

Consider $\mathrm{105\:\div\:5\:=}$

Here dividend =105

divisor = 5

quotient = 21

remainder =0

Euclid’s division algorithm

Euclid’s division algorithm is based on Euclid’s division lemma. Lemmas are the proven statement used to prove another statement. Further moving to Euclid’s division algorithm first we need to understand Euclid’s division lemma.

According to Euclid’s division lemma, for every positive integer 𝑎 & b, there exists unique integers q & r satisfying $\mathrm{a\:=\:bq\:+\:r,\:0\:\leq\:r\:<\:b}$

In VII standard you have studied this lemma as,

$\mathrm{Dividend\:=\:Divisor\:\times\:Quotient\:+\:Reminder}$

Here integer q & r are quotient & remainder

Let's understand Euclid’s division lemma with an example,

When we divided 46 by 5 we get quotient 9 & remainder 1, then by Euclid’s division lemma,

$$\mathrm{39\:=\:5\:\times\:7\:\div\:4}$$

The algorithm is a series of well-defined steps which gives the procedure for solving a type of problem. Euclid’s division algorithm is used to compute the highest common factor (HCF) of the given two integers.

Let’s obtain HCF for two positive integers c & d by using the following steps −

Step 1 − Apply Euclid’s division lemma, so we can find q & r, such that $\mathrm{c\:=\:dq\:+\:r,\:0\:\leq\:r\:<\:d}$ .

Step 2 − If $\mathrm{r\:=\:0}$, then the HCF of c & d is d. This lemma is applicable when $\mathrm{r\:\neq\:0}$.

Step 3 − Continue this process till the remainder is zero. The divisor at the stage will be the required HCF.

Reminder

After performing division, the value that remains or is left out is known as the remainder. When the dividend is not completely divisible by the divisor it left a number or the value is known as the remainder. In mathematics, residual is something that is leftover or remains after calculations have been completed. These remainders are sometimes neglected or round off for obtaining answers in the whole number in many situations.

Properties of the remainder in numeral division:

• The remainder is always less than the divisor

• If the dividend is completely divisible by the divisor completely, then the remainder is zero.

Properties of remainder in polynomial division:

• In polynomial division, the remainder can be found by using the remainder theorem & factor theorem.

• Degree of polynomial should be 1

• Degree of the remainder is always less than the degree of the divisor

• When any polynomial is divided by a linear polynomial with degree 1, the remainder must be constant.

Remainder Theorem in polynomials

This theorem is used to find the remainder when a polynomial is divided by a linear polynomial. When we performed division, the leftover number or term is known as the remainder. So let's discuss the remainder theorem.

Remainder theorem Let $\mathrm{p(x)}$ be any polynomial having a degree greater than or equal to 1 & let be any real number.

Suppose $\mathrm{p(x)}$ is divided by $\mathrm{x\:-\:a}$ , quotient $\mathrm{q(x)}$ & the remainder is $\mathrm{r(x)}$ , then the remainder is $\mathrm{p(a)}$

Proof − Let $\mathrm{p(x)}$ be any polynomial having a degree greater than or equal to 1 & let be any real number. Suppose $\mathrm{p(x)}$ is divided by 𝑥 − 𝑎 , quotient $\mathrm{q(x)}$ & the remainder is $\mathrm{r(x)}$. Mathematically this can be represented as

$$\mathrm{P(x)\:=\:(x\:-\:a)\:\times\:q(x)\:+\:r(x)}$$

Here is the degree of 𝑥 − 𝑎 is one & degree of 𝑟(𝑥) less than the degree of $\mathrm{x\:-\:a}$

Therefore degree of $\mathrm{r(x)\:=\:0}$ . It means 𝑟(𝑥) is constant, say r

$$\mathrm{P(x)\:=\:(x\:-\:a)\:\times\:q(x)\:+\:r}$$

In particular, if we consider 𝑥 = 𝑎 , this equation will give us

$$\mathrm{P(a)\:=\:(a\:-\:a)\:\times\:q(a)\:+\:r}$$

Hence proved

Conclusion

This tutorial covers topics division, Euclid’s division algorithm, remainder, properties of remainder in numeral division & polynomial division & remainder theorem. The division is one of the basic arithmetic operation in mathematics. When a dividend is not completely divisible by the dividend it left a number or value known as the remainder. In mathematics, residual is something that is leftover or remains after calculations have been completed. Euclid’s division algorithm is used to find the highest common factor (HCF) when two integers are given. The remainder theorem is used to find the remainder when the polynomial is divided by a linear polynomial.

FAQs

1. Is zero a reminder?

Yes. When the dividend is completely divisible by the divisor then we get the remainder as zero.

2. State the application of the remainder theorem?

The main application of the remainder theorem is to prove the factor theorem & find the remainder when the polynomial is divided by the linear polynomial.

3. State whether the following statement is true or false. Division by zero is undefined?

True - If any number is multiplied by zero, the answer is zero & reverse it i.e. $\mathrm{}\frac{1}{10}$. It will have infinite value. Therefore, we can’t specify the value in mathematics.

4. Are Euclid’s division lemma & Euclid’s division algorithm the same or not?

Euclid’s division lemma is a proven statement which is used to prove another statement. Whereas Euclid’s division algorithm gives a series of well-defined steps to solve the type of problem.

5. What are the applications of Euclid’s division lemma?

Applications of Euclid’s division lemma −

• It is used for the division of integers.

• It is used to determine the HCF of the positive integers.

• It is used to find properties like odd numbers, even numbers, cube numbers,square numbers etc