Question

The statement \[\text{dividend}=\text{divisor}\times \text{quotient}+\text{remainder }\]is called:
A. Euclid’s multiplication Lemma
B. Euclid’s addition Lemma
C. Euclid’s subtraction Lemma
D. Euclid’s division Lemma

Answer 1

Hint: For this question, we will first define each term from the given statement that is \[\text{dividend}=\text{divisor}\times \text{quotient}+\text{remainder }\] then we will compare it with the given lemma and then compare it with the given lemma and hence get the answer.

Complete step by step answer:
First of all, let’s take each term from the given equation in the question \[\text{dividend}=\text{divisor}\times \text{quotient}+\text{remainder }\]and define each term from it.
So, in arithmetic operations, when we perform the division method, we observe four related terms, they are dividend, divisor, quotient, and remainder.
First of all we have a dividend, the dividend is the value or the amount which we need to divide. It is the whole which is to be divided into different equal parts. For example, if $10$ is divided by $2$ , then the answer will be two equal parts of number $5$ and $10$ is the dividend here.
Now, we have with us the next term that is divisor, now divisor means a number which divides another number. As we saw in above example that $10$ is divided by $2$, here $2$ is the divisor.
Now, we know that the division frequently is shown in algebra by putting the dividend over the divisor with a horizontal line between them. This horizontal line is also called a fraction bar. For example, $m$ divided by $n$ can be represented as $\dfrac{m}{n}$ and this can be read as divide “ $m$ by $n$ ” or “$m$ over $n$” . Here, $m$ is the dividend and $n$ is the divisor. For example, let us take the fraction $\dfrac{5}{6}$. In this fraction, $5$ is the dividend and $6$ is a divisor. The dividend is known as a numerator, and the divisor is known as the denominator in fractions. When the dividend is divided by a divisor, we get a result in either integer form or decimal form.
Now, let’s define quotient and remainder, the quotient is the result obtained from the division process and the number left over after division process is known as the remainder.
Consider an example: $\dfrac{17}{2}$ ,
Let’s divide the given number:
 $2\overset{8}{\overline{\left){\begin{align}
  & 17 \\
 & \underline{16} \\
 & 01 \\
\end{align}}\right.}}$
Here, the dividend is $17$ , divisor is $2$ , quotient is $8$ and the remainder is $1$.
Now, we are given the question the equation: \[\text{dividend}=\text{divisor}\times \text{quotient}+\text{remainder }\] , now this statement is given on the basis of Euclid’s Division Lemma.
Now, according to Euclid’s Division Lemma if we have two positive integers \[a\] and $b$ , then there exist unique integers $q$ and $r$ which satisfies the condition $a=bq+r$ where $0\le r
Hence, the correct option is D.

Note:
The basis of the Euclidean division algorithm is Euclid’s division lemma. To calculate the Highest Common Factor (HCF) of two positive integers $a$ and $b$ we use Euclid’s division algorithm. HCF is the largest number which exactly divides two or more positive integers. That means, on dividing both the integers a and b the remainder is zero.