# The quotient and remainder of the binary division \[{{\left( 101110 \right)}_{2}}\div {{\left( 110 \right)}_{2}}\] respectively are

[a] ${{\left( 111 \right)}_{2}}$ and ${{\left( 100 \right)}_{2}}$

[b] ${{\left( 100 \right)}_{2}}$ and ${{\left( 111 \right)}_{2}}$

[c] ${{\left( 101 \right)}_{2}}$ and ${{\left( 101 \right)}_{2}}$

[d] ${{\left( 100 \right)}_{2}}$ and ${{\left( 100 \right)}_{2}}$

Answer

Verified

378.9k+ views

Hint: Convert the binary numbers in decimal form. Divide to find quotient and remainder. Then convert the remainder and quotient back in binary form to get the answer.

Complete step-by-step answer:

Binary system: In the binary system, every number is represented as a string of 0’s and 1’s. The place value of each place increases two times moving right to the left, e.g. ${{\left( 111 \right)}_{2}}$. The first 1 to the left has place value 4, the second 1 from left has place value 2, and the last one has place value 1.

Decimal system: In the Decimal system, every number is represented using the digits 0-9. The place value of each place increases ten times moving right to the left, e.g. ${{\left( 121 \right)}_{10}}$. The first 1 to the left has place value 100, digit 2 has place value $2\times 10$ , and the last one has place value 1.

Procedure for converting Binary to Decimal:

Find the place value of each digit in the number. The place value \[Digit\times {{2}^{\text{Number of digits to the right of the digit till ''}\centerdot \text{'' }}}\]

e.g., in the number (111011) in binary representation, the place of highlighted 1 = $1\times {{2}^{3}}={{2}^{3}}=8$.

Add the place values of each digit to get the number in decimal representation.

Converting Decimal to Binary.

.Divide P by 2 and note the quotient and remainder.

If quotient = 0, Stop the process

Otherwise, set P = quotient and repeat the above process.

Write all remainders in reverse order.

This is the binary representation of the number.

The above process will become clear in the following example.

Let P = 72

Dividing P by 2 gives quotient = 36 and remainder = 0

So we set P = 36

Dividing P by 2 gives quotient = 18 and remainder = 0

So wet set P = 18

Dividing P by 2 gives quotient = 9 and remainder = 0

So, we set P = 9

Dividing P by 2 gives quotient = 4 and remainder = 1

So, we set P =4

Dividing P by 2 gives quotient = 2 and remainder = 0.

We set P =2

Dividing P by 2 gives quotient = 1 and remainder = 0

We set P = 1

Dividing P by 2 gives quotient = 0 and remainder = 1.

Since quotient = 0 we stop

Writing remainders in reverse order gives 72 = (1001000) in binary representation

Converting 101110 in decimal representation

$\begin{align}

& =1\times {{2}^{5}}+0\times {{2}^{4}}+1\times {{2}^{3}}+1\times {{2}^{2}}+1\times 2+0\times {{2}^{0}} \\

& =32+8+4+2 \\

& =46 \\

\end{align}$

Converting 110 in decimal representation

\[\begin{align}

& =1\times {{2}^{2}}+1\times 2+0\times {{2}^{0}} \\

& =4+2 \\

& =6 \\

\end{align}\]

Dividing 46 by 6 gives quotient = 7 and remainder = 4

Converting 7 in binary representation gives $7\text{ }={{\left( 111 \right)}_{2}}$

Converting 4 in binary representation gives \[4\text{ }={{\left( 100 \right)}_{2}}\]

Hence quotient = ${{\left( 111 \right)}_{2}}$ and remainder = ${{\left( 100 \right)}_{2}}$

Note: [1] We converted the above binary division in decimal form because it is easier to perform division in decimal representation since we are acquainted with it in our day to day life.

[2] The above method for converting decimal to binary is known as an algorithm.

Complete step-by-step answer:

Binary system: In the binary system, every number is represented as a string of 0’s and 1’s. The place value of each place increases two times moving right to the left, e.g. ${{\left( 111 \right)}_{2}}$. The first 1 to the left has place value 4, the second 1 from left has place value 2, and the last one has place value 1.

Decimal system: In the Decimal system, every number is represented using the digits 0-9. The place value of each place increases ten times moving right to the left, e.g. ${{\left( 121 \right)}_{10}}$. The first 1 to the left has place value 100, digit 2 has place value $2\times 10$ , and the last one has place value 1.

Procedure for converting Binary to Decimal:

Find the place value of each digit in the number. The place value \[Digit\times {{2}^{\text{Number of digits to the right of the digit till ''}\centerdot \text{'' }}}\]

e.g., in the number (111011) in binary representation, the place of highlighted 1 = $1\times {{2}^{3}}={{2}^{3}}=8$.

Add the place values of each digit to get the number in decimal representation.

Converting Decimal to Binary.

.Divide P by 2 and note the quotient and remainder.

If quotient = 0, Stop the process

Otherwise, set P = quotient and repeat the above process.

Write all remainders in reverse order.

This is the binary representation of the number.

The above process will become clear in the following example.

Let P = 72

Dividing P by 2 gives quotient = 36 and remainder = 0

So we set P = 36

Dividing P by 2 gives quotient = 18 and remainder = 0

So wet set P = 18

Dividing P by 2 gives quotient = 9 and remainder = 0

So, we set P = 9

Dividing P by 2 gives quotient = 4 and remainder = 1

So, we set P =4

Dividing P by 2 gives quotient = 2 and remainder = 0.

We set P =2

Dividing P by 2 gives quotient = 1 and remainder = 0

We set P = 1

Dividing P by 2 gives quotient = 0 and remainder = 1.

Since quotient = 0 we stop

Writing remainders in reverse order gives 72 = (1001000) in binary representation

Converting 101110 in decimal representation

$\begin{align}

& =1\times {{2}^{5}}+0\times {{2}^{4}}+1\times {{2}^{3}}+1\times {{2}^{2}}+1\times 2+0\times {{2}^{0}} \\

& =32+8+4+2 \\

& =46 \\

\end{align}$

Converting 110 in decimal representation

\[\begin{align}

& =1\times {{2}^{2}}+1\times 2+0\times {{2}^{0}} \\

& =4+2 \\

& =6 \\

\end{align}\]

Dividing 46 by 6 gives quotient = 7 and remainder = 4

Converting 7 in binary representation gives $7\text{ }={{\left( 111 \right)}_{2}}$

Converting 4 in binary representation gives \[4\text{ }={{\left( 100 \right)}_{2}}\]

Hence quotient = ${{\left( 111 \right)}_{2}}$ and remainder = ${{\left( 100 \right)}_{2}}$

Note: [1] We converted the above binary division in decimal form because it is easier to perform division in decimal representation since we are acquainted with it in our day to day life.

[2] The above method for converting decimal to binary is known as an algorithm.

Recently Updated Pages

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

State Gay Lusaaccs law of gaseous volume class 11 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is BLO What is the full form of BLO class 8 social science CBSE

What is pollution? How many types of pollution? Define it

Change the following sentences into negative and interrogative class 10 english CBSE

Which is the tallest animal on the earth A Giraffes class 9 social science CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE