Math Academy

Introduction

We can also use the standard algorithm to divide a four-digit number by a one-digit number.

To illustrate, let's use the algorithm to divide by We start by writing down the question using long division notation:

First, we consider the thousands:

Divide:

Write over thousands:

Multiply:

Subtract:

Bring down:

Next, we consider the hundreds:

Divide:

Write over hundreds:

Multiply:

Subtract:

Bring down:

Next, we consider the tens:

Divide:

Write over tens:

Multiply:

Subtract:

Bring down:

Finally, we consider the ones:

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. The number we have written at the top is the answer. There is no remainder.

Therefore,

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Example: Dividing with No Remainder

A rocket, flying at a constant speed, covered a distance of miles in a period of minutes. How many miles did the rocket cover in minute?

EXPLANATION

To find out how many miles the rocket covered in minute, we need to divide by

We start by writing down the question using long division notation:

Notice that (the number of thousands in ) is less than the divisor Therefore, we should consider both the thousands and hundreds:

Divide:

Write over hundreds:

Multiply:

Subtract:

Bring down:

Next, we consider the tens:

Divide:

Write over tens:

Multiply:

Subtract:

Bring down:

Finally, we consider the ones:

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. The number we have written at the top is the answer. There is no remainder.

Therefore,

So, the rocket covered miles every minute.

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Practice: Dividing with No Remainder

John divides a pumpkin weighing $6,744$ grams into $6$ equal portions. What is the weight of each portion?

a	$924$ pounds
b	$1,124$ pounds
c	$1,214$ pounds
d	$1,142$ pounds
e	$1,421$ pounds

EXPLANATION

To find out what would be the weight of each pumpkin's portion, we need to divide $6,744$ by $6.$

We start by writing down the question using long division notation:

$6$

$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{|}} 7 {\:\phantom{|}} 4 {\:\phantom{|}} 4}$

First, we consider the thousands:

	$\color{blue}1$
$6$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}6} {\:\phantom{\|}} 7 {\:\phantom{\|}} 4 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$6$	$\color{lightgray}\downarrow$
	$0$	$\color{lightgray}7$

Divide: ${\color{red}6} \div 6 = {\color{blue}1}$

Write over thousands: $\color{blue}1$

Multiply: $6 \times {\color{blue}1} = 6$

Subtract: ${\color{red}6} - 6 = 0$

Bring down: $7$

Next, we consider the hundreds:

	$1$	$\color{blue}1$
$6$	$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{\|}} 7 {\:\phantom{\|}} 4 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$6$		$\color{lightgray}\downarrow$
	$0$	$\color{red}7$	$\color{lightgray}\downarrow$
	$-\!\!\!$	$6$	$\color{lightgray}\downarrow$
		$1$	$\color{lightgray}4$

Divide: ${\color{red}7} \div 6 = {\color{blue}1}\,\text{R}\,1$

Write over hundreds: $\color{blue}1$

Multiply: $6 \times {\color{blue}1} = 6$

Subtract: ${\color{red}7} - 6 = 1$

Bring down: $4$

Next, we consider the tens:

	$1$	$1$	$\color{blue}2$
$6$	$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{\|}} 7 {\:\phantom{\|}} 4 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$6$			$\color{lightgray}\downarrow$
	$0$	$7$		$\color{lightgray}\downarrow$
	$-\!\!\!$	$6$		$\color{lightgray}\downarrow$
		$\color{red}1$	$\color{red}4$	$\color{lightgray}\downarrow$
	$-\!\!\!\!\!$	$1$	$2$	$\color{lightgray}\downarrow$
			$2$	$\color{lightgray}4$

Divide: ${\color{red}14} \div 6 = {\color{blue}2}\,\text{R}\,2$

Write over tens: $\color{blue}2$

Multiply: $6 \times {\color{blue}2} = 12$

Subtract: ${\color{red}14} - 12 = 2$

Bring down: $4$

Finally, we consider the ones:

	$1$	$1$	$2$	$\color{blue}4$
$6$	$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{\|}} 7 {\:\phantom{\|}} 4 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$6$
	$0$	$7$
	$-\!\!\!$	$6$
		$1$	$4$
	$-\!\!\!\!\!$	$1$	$2$
			$\color{red}2$	$\color{red}4$
		$-\!\!\!\!\!$	$2$	$4$
				$\fbox{0}$

Divide: ${\color{red}24} \div 6 = {\color{blue}4}$

Write over ones: $\color{blue}4$

Multiply: $6 \times {\color{blue}4} = 24$

Subtract: ${\color{red}24} - 24 = \fbox{0}$

We've gone through all the digits in the number $6,744,$ so the division is done. The number we have written at the top is the answer. There is no remainder.

Therefore, \[ 6,744 \div 6 = 1,124. \]

So, the weight of each portion is $1,124$ grams.

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Practice: Dividing with No Remainder

A rocket covered a distance of $1,274$ miles in $7$ minutes. If the rocket traveled the same distance every minute, how far did the rocket travel every minute?

a	$182$ miles
b	$186$ miles
c	$188$ miles
d	$184$ miles
e	$192$ miles

EXPLANATION

To find out the distance the rocket covered in $1$ minute, we need to divide $1,274$ by $7.$

We start by writing down the question using long division notation:

$7$

$\!\!\require{enclose}\enclose{longdiv}{1 {\:\phantom{|}} 2 {\:\phantom{|}} 7 {\:\phantom{|}} 4}$

Notice that $\color{red}1$ (the number of thousands in ${\color{red}1}274$) is less than the divisor $7.$ Therefore, we should consider both the thousands and hundreds:

		$\color{blue}1$
$7$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}1} {\:\phantom{\|}} {\color{red}2} {\:\phantom{\|}} 7 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$		$7$	$\color{lightgray}\downarrow$
		$5$	$\color{lightgray}7$

Divide: ${\color{red}12} \div 7 = {\color{blue}1}\,\text{R}\,5$

Write over hundreds: $\color{blue}1$

Multiply: $7 \times {\color{blue}1} = 7$

Subtract: ${\color{red}12} - 7 = 5$

Bring down: $7$

Next, we consider the tens:

		$1$	$\color{blue}8$
$7$	$\!\!\require{enclose}\enclose{longdiv}{1 {\:\phantom{\|}} 2 {\:\phantom{\|}} 7 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$		$7$		$\color{lightgray}\downarrow$
		$\color{red}5$	$\color{red}7$	$\color{lightgray}\downarrow$
	$-\!\!\!\!\!$	$5$	$6$	$\color{lightgray}\downarrow$
			$1$	$\color{lightgray}4$

Divide: ${\color{red}57} \div 7 = {\color{blue}8}\,\text{R}\,1$

Write over tens: $\color{blue}8$

Multiply: $7 \times {\color{blue}8} = 56$

Subtract: ${\color{red}57} - 56 = 1$

Bring down: $4$

Finally, we consider the ones:

		$1$	$8$	$\color{blue}2$
$7$	$\!\!\require{enclose}\enclose{longdiv}{1 {\:\phantom{\|}} 2 {\:\phantom{\|}} 7 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$		$7$
		$5$	$7$
	$-\!\!\!\!\!$	$5$	$6$
			$\color{red}1$	$\color{red}4$
		$-\!\!\!\!\!$	$1$	$4$
				$\fbox{0}$

Divide: ${\color{red}14} \div 7 = {\color{blue}2}$

Write over ones: $\color{blue}2$

Multiply: $7 \times {\color{blue}2} = 14$

Subtract: ${\color{red}14} - 14 = \fbox{0}$

We've gone through all the digits in the number $1,274,$ so the division is done. The number we have written at the top is the answer. There is no remainder.

Therefore, \[ 1,274 \div 7 = 182. \]

So, the rocket covered $182$ miles every minute.

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Example: Dividing With a Remainder

What are the quotient and the remainder of dividing by

EXPLANATION

We start by writing down the question using long division notation:

Notice that (the number of thousands in ) is less than the divisor Therefore, we should consider both the thousands and hundreds:

Divide:

Write over hundreds:

Multiply:

Subtract:

Bring down:

Next, we consider the tens:

Divide:

Write over tens:

Notice that is less than the divisor but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

Bring down:

Finally, we consider the ones:

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. However, we still have a remainder of so we include this in our final answer.

Therefore, when we divide by

the quotient is and
the remainder is

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Practice: Dividing With a Remainder

What are the quotient and the remainder of dividing $7,269$ by $4?$

a	The quotient is $1,821,$ and the remainder is $4$
b	The quotient is $1,819,$ and the remainder is $0$
c	The quotient is $1,827,$ and the remainder is $2$
d	The quotient is $1,806,$ and the remainder is $3$
e	The quotient is $1,817,$ and the remainder is $1$

EXPLANATION

We start by writing down the question using long division notation:

$4$

$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{|}} 2 {\:\phantom{|}} 6 {\:\phantom{|}} 9}$

First, we consider the thousands:

	$\color{blue}1$
$4$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}7} {\:\phantom{\|}} 2 {\:\phantom{\|}} 6 {\:\phantom{\|}} 9}$
$-\!\!\!\!\!\!\!$	$4$	$\color{lightgray}\downarrow$
	$3$	$\color{lightgray}2$

Divide: ${\color{red}7} \div 4 = {\color{blue}1}\,\text{R}\,3$

Write over thousands: $\color{blue}1$

Multiply: $4 \times {\color{blue}1} = 4$

Subtract: ${\color{red}7} - 4 = 3$

Bring down: $2$

Next, we consider the hundreds:

	$1$	$\color{blue}8$
$4$	$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{\|}} 2 {\:\phantom{\|}} 6 {\:\phantom{\|}} 9}$
$-\!\!\!\!\!\!\!$	$4$		$\color{lightgray}\downarrow$
	$\color{red}3$	$\color{red}2$	$\color{lightgray}\downarrow$
$-\!\!\!\!\!\!\!$	$3$	$2$	$\color{lightgray}\downarrow$
		$0$	$\color{lightgray}6$

Divide: ${\color{red}32} \div 4 = {\color{blue}8}$

Write over hundreds: $\color{blue}8$

Multiply: $4 \times {\color{blue}8} = 32$

Subtract: ${\color{red}32} - 32 = 0$

Bring down: $6$

Next, we consider the tens:

	$1$	$8$	$\color{blue}1$
$4$	$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{\|}} 2 {\:\phantom{\|}} 6 {\:\phantom{\|}} 9}$
$-\!\!\!\!\!\!\!$	$4$			$\color{lightgray}\downarrow$
	$3$	$2$		$\color{lightgray}\downarrow$
$-\!\!\!\!\!\!\!$	$3$	$2$		$\color{lightgray}\downarrow$
		$0$	$\color{red}6$	$\color{lightgray}\downarrow$
		$-\!\!\!$	$4$	$\color{lightgray}\downarrow$
			$2$	$\color{lightgray}9$

Divide: ${\color{red}6} \div 4 = {\color{blue}1}\,\text{R}\,2$

Write over tens: $\color{blue}1$

Multiply: $4 \times {\color{blue}1} = 4$

Subtract: ${\color{red}6} - 4 = 2$

Bring down: $9$

Finally, we consider the ones:

	$1$	$8$	$1$	$\color{blue}7$
$4$	$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{\|}} 2 {\:\phantom{\|}} 6 {\:\phantom{\|}} 9}$
$-\!\!\!\!\!\!\!$	$4$
	$3$	$2$
$-\!\!\!\!\!\!\!$	$3$	$2$
		$0$	$6$
		$-\!\!\!$	$4$
			$\color{red}2$	$\color{red}9$
		$-\!\!\!\!\!$	$2$	$8$
				$\fbox{1}$

Divide: ${\color{red}29} \div 4 = {\color{blue}7}\,\text{R}\,1$

Write over ones: $\color{blue}7$

Multiply: $4 \times {\color{blue}7} = 28$

Subtract: ${\color{red}29} - 28 = \fbox{1}$

We've gone through all the digits in the number $7,269,$ so the division is done. However, we still have a remainder of $1,$ so we include this in our final answer.

Therefore, when we divide $7,269$ by $4,$

the quotient is $1,817,$ and
the remainder is $1.$

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Practice: Dividing With a Remainder

What are the quotient and the remainder of dividing $7,622$ by $5?$

a	The quotient is $1,518,$ and the remainder is $2$
b	The quotient is $1,532,$ and the remainder is $7$
c	The quotient is $1,524,$ and the remainder is $2$
d	The quotient is $1,526,$ and the remainder is $7$
e	The quotient is $1,520,$ and the remainder is $2$

EXPLANATION

We start by writing down the question using long division notation:

$5$

$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{|}} 6 {\:\phantom{|}} 2 {\:\phantom{|}} 2}$

First, we consider the thousands:

	$\color{blue}1$
$5$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}7} {\:\phantom{\|}} 6 {\:\phantom{\|}} 2 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$5$	$\color{lightgray}\downarrow$
	$2$	$\color{lightgray}6$

Divide: ${\color{red}7} \div 5 = {\color{blue}1}\,\text{R}\,2$

Write over thousands: $\color{blue}1$

Multiply: $5 \times {\color{blue}1} = 5$

Subtract: ${\color{red}7} - 5 = 2$

Bring down: $6$

Next, we consider the hundreds:

	$1$	$\color{blue}5$
$5$	$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{\|}} 6 {\:\phantom{\|}} 2 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$5$		$\color{lightgray}\downarrow$
	$\color{red}2$	$\color{red}6$	$\color{lightgray}\downarrow$
$-\!\!\!\!\!\!\!$	$2$	$5$	$\color{lightgray}\downarrow$
		$1$	$\color{lightgray}2$

Divide: ${\color{red}26} \div 5 = {\color{blue}5}\,\text{R}\,1$

Write over hundreds: $\color{blue}5$

Multiply: $5 \times {\color{blue}5} = 25$

Subtract: ${\color{red}26} - 25 = 1$

Bring down: $2$

Next, we consider the tens:

	$1$	$5$	$\color{blue}2$
$5$	$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{\|}} 6 {\:\phantom{\|}} 2 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$5$			$\color{lightgray}\downarrow$
	$2$	$6$		$\color{lightgray}\downarrow$
$-\!\!\!\!\!\!\!$	$2$	$5$		$\color{lightgray}\downarrow$
		$\color{red}1$	$\color{red}2$	$\color{lightgray}\downarrow$
	$-\!\!\!\!\!$	$1$	$0$	$\color{lightgray}\downarrow$
			$2$	$\color{lightgray}2$

Divide: ${\color{red}12} \div 5 = {\color{blue}2}\,\text{R}\,2$

Write over tens: $\color{blue}2$

Multiply: $5 \times {\color{blue}2} = 10$

Subtract: ${\color{red}12} - 10 = 2$

Bring down: $2$

Finally, we consider the ones:

	$1$	$5$	$2$	$\color{blue}4$
$5$	$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{\|}} 6 {\:\phantom{\|}} 2 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$5$
	$2$	$6$
$-\!\!\!\!\!\!\!$	$2$	$5$
		$1$	$2$
	$-\!\!\!\!\!$	$1$	$0$
			$\color{red}2$	$\color{red}2$
		$-\!\!\!\!\!$	$2$	$0$
				$\fbox{2}$

Divide: ${\color{red}22} \div 5 = {\color{blue}4}\,\text{R}\,2$

Write over ones: $\color{blue}4$

Multiply: $5 \times {\color{blue}4} = 20$

Subtract: ${\color{red}22} - 20 = \fbox{2}$

We've gone through all the digits in the number $7,622,$ so the division is done. However, we still have a remainder of $2,$ so we include this in our final answer.

Therefore, when we divide $7,622$ by $5,$

the quotient is $1,524,$ and
the remainder is $2.$

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Example: Bringing Down More Than One Digit at a Time

Find the value of

EXPLANATION

We start by writing down the question using long division notation:

First, we consider the thousands:

Divide:

Write over thousands:

Multiply:

Subtract:

Bring down:

Next, we consider the hundreds:

Divide:

Write over hundreds:

Notice that is less than the divisor but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

Bring down:

Next, we consider the tens:

Divide:

Write over ones:

Multiply:

Subtract:

Bring down:

Finally, we consider the ones:

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. The number we have written at the top is the answer. There is no remainder.

Therefore,

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Practice: Bringing Down More Than One Digit at a Time

Find the value of $2,114 \div 3.$

a	$706\, \text{R}5$
b	$707\, \text{R}3$
c	$705\, \text{R}9$
d	$703\, \text{R}5$
e	$704\, \text{R}2$

EXPLANATION

We start by writing down the question using long division notation:

$3$

$\!\!\require{enclose}\enclose{longdiv}{2 {\:\phantom{|}} 1 {\:\phantom{|}} 1 {\:\phantom{|}} 4}$

Notice that $\color{red}2$ (the number of thousands in ${\color{red}2},114$) is less than the divisor $8.$ Therefore, we should consider both the thousands and hundreds:

		$\color{blue}7$
$3$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}2} {\:\phantom{\|}} {\color{red}1} {\:\phantom{\|}} 1 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$2$	$1$	$\color{lightgray}\downarrow$
		$0$	$\color{lightgray}1$

Divide: ${\color{red}21} \div 3 = {\color{blue}7}$

Write over hundreds: $\color{blue}7$

Multiply: $3 \times {\color{blue}7} = 21$

Subtract: ${\color{red}21} - 21 = 0$

Bring down: $1$

Next, we consider the tens:

		$7$	$\color{blue}0$
$3$	$\!\!\require{enclose}\enclose{longdiv}{2 {\:\phantom{\|}} 1 {\:\phantom{\|}} 1 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$2$	$1$
		$0$	$\color{red}1$

Divide: ${\color{red}1} \div 3 = {\color{blue}0}\,\text{R}\,1$

Write over tens: $\color{blue}0$

Notice that $\color{red}1$ is less than the divisor $3$ but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

		$7$	$0$
$3$	$\!\!\require{enclose}\enclose{longdiv}{2 {\:\phantom{\|}} 1 {\:\phantom{\|}} 1 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$2$	$1$		$\color{lightgray}\downarrow$
		$0$	$1$	$\color{lightgray}4$

Bring down: $4$

Finally, we consider the ones:

		$7$	$0$	$\color{blue}4$
$3$	$\!\!\require{enclose}\enclose{longdiv}{2 {\:\phantom{\|}} 1 {\:\phantom{\|}} 1 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$2$	$1$
		$0$	$\color{red}1$	$\color{red}4$
		$-\!\!\!\!\!$	$1$	$2$
				$\fbox{2}$

Divide: ${\color{red}14} \div 3 = {\color{blue}4}\,\text{R}\,2$

Write over ones: $\color{blue}4$

Multiply: $3 \times {\color{blue}4} = 12$

Subtract: ${\color{red}14} - 12 = \fbox{2}$

We've gone through all the digits in the number $2,114$ so the division is done. However, we still have a remainder of $2,$ so we include this in our final answer.

Therefore, \[ 2,114 \div 3 = 704\, \text{R}2. \]

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Practice: Bringing Down More Than One Digit at a Time

Find the value of $1,324 \div 4.$

a	$331$
b	$336$
c	$327$
d	$317$
e	$341$

EXPLANATION

We start by writing down the question using long division notation:

$4$

$\!\!\require{enclose}\enclose{longdiv}{1 {\:\phantom{|}} 3 {\:\phantom{|}} 2 {\:\phantom{|}} 4}$

Notice that $\color{red}1$ (the number of thousands in ${\color{red}1},324$) is less than the divisor $4.$ Therefore, we should consider both the thousands and hundreds:

		$\color{blue}3$
$4$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}1} {\:\phantom{\|}} {\color{red}3} {\:\phantom{\|}} 2 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$1$	$2$	$\color{lightgray}\downarrow$
		$1$	$\color{lightgray}2$

Divide: ${\color{red}13} \div 4 = {\color{blue}3}\,\text{R}\,1$

Write over hundreds: $\color{blue}3$

Multiply: $4 \times {\color{blue}3} = 12$

Subtract: ${\color{red}13} - 12 = 1$

Bring down: $2$

Next, we consider the tens:

		$3$	$\color{blue}3$
$4$	$\!\!\require{enclose}\enclose{longdiv}{1 {\:\phantom{\|}} 3 {\:\phantom{\|}} 2 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$1$	$2$		$\color{lightgray}\downarrow$
		$\color{red}1$	$\color{red}2$	$\color{lightgray}\downarrow$
	$-\!\!\!\!\!$	$1$	$2$	$\color{lightgray}\downarrow$
			$0$	$\color{lightgray}4$

Divide: ${\color{red}12} \div 4 = {\color{blue}3}$

Write over tens: $\color{blue}3$

Multiply: $4 \times {\color{blue}3} = 12$

Subtract: ${\color{red}12} - 12 = 0$

Bring down: $4$

Finally, we consider the ones:

		$3$	$3$	$\color{blue}1$
$4$	$\!\!\require{enclose}\enclose{longdiv}{1 {\:\phantom{\|}} 3 {\:\phantom{\|}} 2 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$1$	$2$
		$1$	$2$
	$-\!\!\!\!\!$	$1$	$2$
			$0$	$\color{red}4$
			$-\!\!\!\!\!$	$4$
				$\fbox{0}$

Divide: ${\color{red}4} \div 4 = {\color{blue}1}$

Write over ones: $\color{blue}1$

Multiply: $4 \times {\color{blue}1} = 4$

Subtract: ${\color{red}4} - 4 = \fbox{0}$

We've gone through all the digits in the number $1,324$ so the division is done. The number we have written at the top is the answer. There is no remainder.

Therefore, \[ 1,324 \div 4 = 331. \]

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Example: Bringing Down More Than Two Digits at a Time

Find the quotient and the remainder of divided by

EXPLANATION

We start by writing down the question using long division notation:

First, we consider the thousands:

Divide:

Write over thousands:

Multiply:

Subtract:

Bring down:

Next, we consider the hundreds:

Divide:

Write over hundreds:

Notice that is less than the divisor but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

Bring down:

Next, we consider the tens:

Divide:

Write over hundreds:

Notice that is less than the divisor but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

Bring down:

Finally, we consider both tens and ones:

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. However, we still have a remainder of so we include this in our final answer.

Therefore,

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Practice: Bringing Down More Than Two Digits at a Time

Find the quotient and the remainder of $6,035$ divided by $6.$

a	$1,003\,\text{R}\,5$
b	$1,025\,\text{R}\,0$
c	$1,005\,\text{R}\,5$
d	$1,006\,\text{R}\,5$
e	$1,055\,\text{R}\,5$

EXPLANATION

We start by writing down the question using long division notation:

$6$

$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{|}} 0 {\:\phantom{|}} 3 {\:\phantom{|}} 5}$

First, we consider the thousands:

	$\color{blue}1$
$6$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}6} {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$6$	$\color{lightgray}\downarrow$
	$0$	$\color{lightgray}0$

Divide: ${\color{red}6} \div 6 = {\color{blue}1}$

Write over thousands: $\color{blue}1$

Multiply: $6 \times {\color{blue}1} = 6$

Subtract: ${\color{red}6} - 6 = 0$

Bring down: $0$

Next, we consider the hundreds:

	$1$	$\color{blue}0$
$6$	$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$6$
	$0$	$\color{red}0$

Divide: ${\color{red}0} \div 6 = {\color{blue}0}$

Write over hundreds: $\color{blue}0$

Notice that $\color{red}0$ is less than the divisor $6$ but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

	$1$	$0$
$6$	$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$6$		$\color{lightgray}\downarrow$
	$0$	$0$	$\color{lightgray}3$

Bring down: $3$

Next, we consider the tens:

	$1$	$0$	$\color{blue}0$
$6$	$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$6$
	$0$	$0$	$\color{red}3$

Divide: ${\color{red}3} \div 6 = {\color{blue}0}\,\text{R}\,3$

Write over tens: $\color{blue}0$

Notice that $\color{red}3$ is less than the divisor $6$ but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

	$1$	$0$	$0$
$6$	$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$6$			$\color{lightgray}\downarrow$
	$0$	$0$	$3$	$\color{lightgray}5$

Bring down: $5$

Finally, we consider both tens and ones:

	$1$	$0$	$0$	$\color{blue}5$
$6$	$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$6$
	$0$	$0$	$\color{red}3$	$\color{red}5$
		$-\!\!\!\!\!$	$3$	$0$
				$\fbox{5}$

Divide: ${\color{red}35} \div 6 = {\color{blue}5}\,\text{R}\,5$

Write over ones: $\color{blue}5$

Multiply: $6 \times {\color{blue}5} = 30$

Subtract: ${\color{red}35} - 30 = \fbox{5}$

We've gone through all the digits in the number $6,035,$ so the division is done. However, we still have a remainder of $5,$ so we include this in our final answer.

Therefore, \[ 6,035 \div 6 = 1,005\,\text{R}\,5. \]

FLAG

Practice: Bringing Down More Than Two Digits at a Time

Find the quotient and the remainder of $4,032$ divided by $4 .$

a	$1,103$
b	$1,013$
c	$1,009$
d	$1,008$
e	$1,118$

EXPLANATION

We start by writing down the question using long division notation:

$4$

$\!\!\require{enclose}\enclose{longdiv}{4 {\:\phantom{|}} 0 {\:\phantom{|}} 3 {\:\phantom{|}} 2}$

First, we consider the thousands:

	$\color{blue}1$
$4$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}4} {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$4$	$\color{lightgray}\downarrow$
	$0$	$\color{lightgray}0$

Divide: ${\color{red}4} \div 4 = {\color{blue}1}$

Write over thousands: $\color{blue}1$

Multiply: $4 \times {\color{blue}1} = 4$

Subtract: ${\color{red}4} - 4 = 0$

Bring down: $0$

Next, we consider the hundreds:

	$1$	$\color{blue}0$
$4$	$\!\!\require{enclose}\enclose{longdiv}{4 {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$4$
	$0$	$\color{red}0$

Divide: ${\color{red}0} \div 4 = {\color{blue}0}$

Write over hundreds: $\color{blue}0$

Notice that $\color{red}0$ is less than the divisor $4$ but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

	$1$	$0$
$4$	$\!\!\require{enclose}\enclose{longdiv}{4 {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$4$		$\color{lightgray}\downarrow$
	$0$	$0$	$\color{lightgray}3$

Bring down: $3$

Next, we consider the tens:

	$1$	$0$	$\color{blue}0$
$4$	$\!\!\require{enclose}\enclose{longdiv}{4 {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$4$
	$0$	$0$	$\color{red}3$

Divide: ${\color{red}3} \div 4 = {\color{blue}0}\,\text{R}\,3$

Write over tens: $\color{blue}0$

Notice that $\color{red}3$ is less than the divisor $4$ but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

	$1$	$0$	$0$
$4$	$\!\!\require{enclose}\enclose{longdiv}{4 {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$4$			$\color{lightgray}\downarrow$
	$0$	$0$	$3$	$\color{lightgray}2$

Bring down: $2$

Finally, we consider both tens and ones:

	$1$	$0$	$0$	$\color{blue}8$
$4$	$\!\!\require{enclose}\enclose{longdiv}{4 {\:\phantom{\|}} 0 {\:\phantom{\|}} 3 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$4$
	$0$	$0$	$\color{red}3$	$\color{red}2$
		$-\!\!\!\!\!$	$3$	$2$
				$\fbox{0}$

Divide: ${\color{red}32} \div 4 = {\color{blue}8}$

Write over ones: $\color{blue}8$

Multiply: $4 \times {\color{blue}8} = 32$

Subtract: ${\color{red}32} - 32 = \fbox{0}$

We've gone through all the digits in the number $4,032,$ so the division is done.

Therefore, \[ 4,032 \div 4 = 1,008. \]

FLAG