2.3: Division of Whole Numbers

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Learning Objectives

be able to divide a whole number by a single or multiple digit divisor
be able to interpret a calculator statement that a division results in a remainder

Division with a Single Digit Divisor

Our experience with multiplication of whole numbers allows us to perform such divisions as $75 \div 5$. We perform the division by performing the corresponding multiplication, $5 \times Q = 75$. Each division we considered in [link] had a one-digit quotient. Now we will consider divisions in which the quotient may consist of two or more digits. For example, $75 \div 5$.

Let's examine the division $75 \div 5$. We are asked to determine how many 5's are contained in 75. We'll approach the problem in the following way.

Make an educated guess based on experience with multiplication.
Find how close the estimate is by multiplying the estimate by 5.
If the product obtained in step 2 is less than 75, find out how much less by subtracting it from 75.
If the product obtained in step 2 is greater than 75, decrease the estimate until the product is less than 75. Decreasing the estimate makes sense because we do not wish to exceed 75.

We can suggest from this discussion that the process of division consists of

The Four Steps in Division

an educated guess
a multiplication
a subtraction
bringing down the next digit (if necessary)

The educated guess can be made by determining how many times the divisor is contained in the dividend by using only one or two digits of the dividend.

Sample Set A

Find $75 \div 5$.

Solution

$5\overline{)75}$ Rewrite the problem using a division bracket.

$\begin{array} {r} {10} \\ {5\overline{)75}} \end{array}$

Make an educated guess by noting that one 5 is contained in 75 at most 10 times.
Since 7 is the tens digit, we estimate that 5 goes into 75 at most 10 times.

$\begin{array} {r} {10} \\ {5\overline{)75}} \\ {\underline{-50}} \\ {25} \end{array}$

Now determine how close the estimate is.
10 fives is $10 \times 5 = 50$. Subtract 50 from 75.
Estimate the number of 5's in 25.
There are exactly 5 fives in 25.

$屏幕快照 2020-09-29 上午10.20.37.png$

Check:

$Is 75 equal to 15 times 5? Yes.$

Thus, $75 \div 5 = 15$.

The notation in this division can be shortened by writing.

$\begin{array} {r} {15} \\ {5\overline{)75}} \\ {-5 \downarrow} \\ {\overline{\ \ 25}} \\ {-25} \\ {\overline{\ \ \ \ 0}} \end{array}$

$\begin{cases} \text{Divide:} & \text{5 goes into 7 at most 1 time.} \\ \text{Multiply:} & 1 \times 5 = 5. \text{Write 5 below 7.} \\ \text{Subtract:} & 7 - 5 = 2. \text{Bring down the 5.} \end{cases}$ $\begin{cases} {\text{Divide:}} & {\text{5 goes into 25 exactly 5 times}} \\ {\text{Multiply:}} & {5 \times 5 = 25. \text{Write 25 below 25.}} \\ {\text{Subtract:}} & {25 - 25 = 0} \end{cases}$

Sample Set A

Find $4,944 \div 8$.

Solution

$8\overline{)4944}$

Rewrite the problem using a division bracket.

$\begin{array} {r} {600} \\ {8\overline{)4944}} \\ {\underline{-4800}} \\ {144} \end{array}$

8 goes into 49 at most 6 times, and 9 is in the hundreds column. We'll guess 600.
Then, $8 \times 600 = 4800$.

$\begin{array} {r} {10} \\ {600} \\ {8\overline{)4944}} \\ {\underline{-4800}} \\ {144} \\ {\underline{-\ \ 80}} \\ {64} \end{array}$

8 goes into 14 at most 1 time, and 4 is in the tens column. We'll guess 10.

$\begin{array} {r} {8} \\ {10} \\ {600} \\ {8\overline{)4944}} \\ {\underline{-4800}} \\ {144} \\ {\underline{-\ \ 80}} \\ {64} \\ {\underline{-64}} \\ {0} \end{array}$

8 goes into 64 exactly 8 times.
600 eights + 10 eights + 8 eights = 618 eights.

Check:

$Is 4944 equal to 8 times 18? Yes.$

Thus, $4,944 \div 8 = 618$.

As in the first problem, the notation in this division can be shortened by eliminating the subtraction signs and the zeros in each educated guess.

$Long division. 4944 divided by 8. After each educated guess, the digit to the right is brought down to the next line.$

$\begin{cases} \text{Divide:} & \text{8 goes into 49 at most 6 times.} \\ \text{Multiply:} & 6 \times 8 = 48. \text{Write 48 below 49.} \\ \text{Subtract:} & 49 - 48 = 1. \text{Bring down the 4.} \end{cases}$ $\begin{cases} \text{Divide:} & \text{8 goes into 14 at most 1 time.} \\ \text{Multiply:} & 1 \times 8 = 8. \text{Write 8 below 14.} \\ \text{Subtract:} & 14 - 8 = 6. \text{Bring down the 4.} \end{cases}$

$\begin{cases} \text{Divide:} & \text{8 goes into 64 exactly 8 times.} \\ \text{Multiply:} & 8 \times 8 = 64. \text{Write 64 below 64.} \\ \text{Subtract:} & 64 - 64 = 0. \end{cases}$

Note

Not all divisions end in zero. We will examine such divisions in a subsequent subsection.

Practice Set A

Perform the following divisions.

$126 \div 7$

Answer: 18

Practice Set A

$324 \div 4$

Answer: 81

Practice Set A

$2,559 \div 3$

Answer: 853

Practice Set A

$5,645 \div 5$

Answer: 1,129

Practice Set A

$757,125 \div 9$

Answer: 84,125

Division with a Multiple Digit Divisor

The process of division also works when the divisor consists of two or more digits. We now make educated guesses using the first digit of the divisor and one or two digits of the dividend.

Sample Set B

Find $2,232 \div 36$.

Solution

$36 \overline{)2232}$

Use the first digit of the divisor and the first two digits of the dividend to make the educated guess.

3 goes into 22 at most 7 times.

Try 7: $7 \times 36 = 252$ which is greater than 223. Reduce the estimate.

Try 6: $6 \times 36 = 216$ which is less than 223.

$The first step of a long division problem. 2232 divided by 36. 36 goes into 223 approximately 6 times, with a remainder of 7. The ones digit of 2232 is then brought down to adjoin the 7.$

$\begin{array} {ll} {\text{Multiply: }} & {6 \times 36 = 216. \text{Write 216 below 223.}} \\ {\text{Subtract: }} & {223 - 216 = 7. \text{Bring down the 2.}} \end{array}$

Divide 3 into 7 to estimate the number of times 36 goes into 72. The 3 goes into 7 at most 2 times.

Try 2: $2 \times 36 = 72$.

$Long division. 2232 divided by 36. 36 goes into 223 approximately 6 times, with a remainder of 7. The ones digit of 2232 is then brought down to adjoin the 7. 36 goes into 72 exactly twice, leaving a remainder of 0.$

Check:

$Is 2232 equal to 36 time 62? Yes.$

Thus $2,232 \div 36 = 62$.

Sample Set B

Find $2,417,228 \div 802$.

Solution

$802 \overline{)2417228}$

First, the educated guess: $24 \div 8 = 3$. Then $3 \times 802 = 2406$, which is less than 2417. Use 3 as the guess. Since $3 \times 802 = 2406$, and 2406 has four digits, place the 3 above the fourth digit of the dividend.

$The first step of a long division problem. 2417228 divided by 802. 802 goes into 2417 approximately 3 times, with a remainder of 11. The hundreds digit of 2417228 is then brought down to adjoin the 11.$

Subtract: 2417 - 2406 = 11.
Bring down the 2.

The divisor 802 goes into 112 at most 0 times. Use 0.

$The second step of a long division problem. 802 goes into 112 0 times, so a zero is placed above, and the next digit is brought down.$

$\begin{array} {ll} {\text{Multiply:}} & {0 \times 802 = 0.} \\ {\text{Subtract:}} & {112 - 0 = 112.} \\ {\text{Bring down the 2.}} & {} \end{array}$

The 8 goes into 11 at most 1 time, and $1 \times 802 = 802$, which is less than 1122. Try 1.

$The third step of a long division problem. 802 goes into 1122 once, so a 1 is placed above and the ones digit is brought down.$

Subtract 1122 - 802 = 320
Bring down the 8.

8 goes into 32 at most 4 times.

$4 \times 802 = 3208$.

Use 4.

$The third step of a long division problem. 2417228 divided by 802. 802 goes into 2417 approximately 3 times, with a remainder of 11. The hundreds digit of 2417228 is then brought down to adjoin the 11. 802 goes into 112 0 times, so a zero is placed above, and the next digit is brought down. 802 goes into 1122 once, so a 1 is placed above and the ones digit is brought down. 802 goes into 3208 4 times, leaving a remainder of 0.$

Check:

$Is 2417228 equal to 3014 times 802? Yes.$

Thus, $2,417,228 \div 802 = 3,014$.

Practice Set B

Perform the following divisions.

$1,376 \div 32$

Answer: 43

Practice Set B

$6,160 \div 55$

Answer: 112

Practice Set B

$18,605 \div 61$

Answer: 305

Practice Set B

$144,768 \div 48$

Answer: 3,016

Division with a Remainder

We might wonder how many times 4 is contained in 10. Repeated subtraction yields

$\begin{array} {r} {10} \\ {\underline{-\ \ 4}} \\ {6} \\ {\underline{-4}} \\ {2} \end{array}$

Since the remainder is less than 4, we stop the subtraction. Thus, 4 goes into 10 two times with 2 remaining. We can write this as a division as follows.

$\begin{array} {r} {2} \\ {4 \overline{)10}} \\ {\underline{-\ \ 8}} \\ {2} \end{array}$

$\begin{array} {ll} {\text{Divide:}} & {\text{4 goes into 10 at most 2 times.}} \\ {\text{Multiply:}} & {2 \times 4 = 8. \text{Write 8 below 0.}} \\ {\text{Subtract:}} & {10 - 8 = 2.} \end{array}$

$\begin{array} {r} {\text{2R2}} \\ {4\overline{)\ \ \ 10}} \\ {\underline{-8}} \\ {2} \end{array}$ or $10 \div 4 = \begin{matrix} \underbrace{\text{2R2}} \\ {\text{2 with remainder 2}} \end{matrix}$

Sample Set C

Find $85 \div 3$.

Solution

$Long division. 85 divided by 3. 3 goes into 8 twice, with a remainder of 2. The ones digit is then brought down. 3 goes into 25 8 times, with a remainder of 1.$

$\begin{cases} \text{Divide:} & \text{3 goes into 8 at most 2 times.} \\ \text{Multiply:} & 2 \times 3 = 6. \text{ Write 6 below 8.} \\ \text{Subtract:} & 8 - 6 = 2. \text{ Bring down the 5.} \end{cases}$ $\begin{cases} \text{Divide:} & \text{3 goes into 25 at most 8 times.} \\ \text{Multiply:} & 3 \times 8 = 24. \text{ Write 24 below 25.} \\ \text{Subtract:} & 25 - 24 = 1 \end{cases}$

Sample Set C

Find $726 \div 23$.

Solution

$Long division. 726 divided by 23. 23 goes into 72 three times, with a remainder of 3. The ones digit is then brought down. 23 goes into 36 once, with a remainder of 13.$

Check: Multiply 31 by 23, then add 13.

$31 times 23 equals 713. 713 plus 13 equals 726.$

Thus, $726 \div 23 = 31R13$.

Practice Set C

Perform the following divisions.

$75 \div 4$

Answer: 18 R3

Practice Set C

$346 \div 8$

Answer: 43 R2

Practice Set C

$489 \div 21$

Answer: 23 R6

Practice Set C

$5,016 \div 82$

Answer: 61 R14

Practice Set C

$41,196 \div 67$

Answer: 614 R58

Calculators

The calculator can be useful for finding quotients with single and multiple digit divisors. If, however, the division should result in a remainder, the calculator is unable to provide us with the particular value of the remainder. Also, some calculators (most nonscientific) are unable to perform divisions in which one of the numbers has more than eight digits.

Sample Set D

Use a calculator to perform each division.

$328 \div 8$.

Solution

Type	328
Press	$\div$
Type	8
Press	=

The display now reads 41.

Sample Set D

$53,136 \div 82$.

Solution

Type	53136
Press	$\div$
Type	82
Press	=

The display now reads 648.

Sample Set D

$730,019,001 \div 326$

Solution

We first try to enter 730,019,001 but find that we can only enter 73001900. If our calculator has only an eight-digit display (as most nonscientific calculators do), we will be unable to use the calculator to perform this division.

Sample Set D

$3727 \div 49$.

Solution

Type	3727
Press	$\div$
Type	49
Press	=

The display now reads 76.061224.

This number is an example of a decimal number (see [link]). When a decimal number results in a calculator division, we can conclude that the division produces a remainder.

Practice Set D

Use a calculator to perform each division.

$3,330 \div 74$

Answer: 45

Practice Set D

$63,365 \div 115$

Answer: 551

Practice Set D

$21,996,385,287 \div 53$

Answer: Since the dividend has more than eight digits, this division cannot be performed on most nonscientific calculators. On others, the answer is 415,026,137.4

Practice Set D

$4,558 \div 67$

Answer: This division results in 68.02985075, a decimal number, and therefore, we cannot, at this time, find the value of the remainder. Later, we will discuss decimal numbers.

Exercises

For the following problems, perform the divisions.

The first 38 problems can be checked with a calculator by multiplying the divisor and quotient then adding the remainder.

Exercise $\PageIndex{1}$

$52 \div 4$

Answer: 13

Exercise $\PageIndex{2}$

$776 \div 8$

Exercise $\PageIndex{3}$

$603 \div 9$

Answer: 67

Exercise $\PageIndex{4}$

$240 \div 8$

Exercise $\PageIndex{5}$

$208 \div 4$

Answer: 52

Exercise $\PageIndex{6}$

$576 \div 6$

Exercise $\PageIndex{7}$

$21 \div 7$

Answer: 3

Exercise $\PageIndex{8}$

$0 \div 0$

Exercise $\PageIndex{9}$

$140 \div 2$

Answer: 70

Exercise $\PageIndex{10}$

$528 \div 8$

Exercise $\PageIndex{11}$

$244 \div 4$

Answer: 61

Exercise $\PageIndex{12}$

$0 \div 7$

Exercise $\PageIndex{13}$

$177 \div 3$

Answer: 59

Exercise $\PageIndex{14}$

$96 \div 8$

Exercise $\PageIndex{15}$

$67 \div 1$

Answer: 67

Exercise $\PageIndex{16}$

$896 \div 56$

Exercise $\PageIndex{17}$

$1,044 \div 12$

Answer: 87

Exercise $\PageIndex{18}$

$988 \div 19$

Exercise $\PageIndex{19}$

$5,238 \div 97$

Answer: 54

Exercise $\PageIndex{20}$

$2530 \div 55$

Exercise $\PageIndex{21}$

$4,264 \div 82$

Answer: 52

Exercise $\PageIndex{22}$

$637 \div 13$

Exercise $\PageIndex{23}$

$3,420 \div 90$

Answer: 38

Exercise $\PageIndex{24}$

$5,655 \div 87$

Exercise $\PageIndex{25}$

$2,115 \div 47$

Answer: 45

Exercise $\PageIndex{26}$

$9,328 \div 22$

Exercise $\PageIndex{27}$

$55,167 \div 71$

Answer: 777

Exercise $\PageIndex{28}$

$68,356 \div 92$

Exercise $\PageIndex{29}$

$27,702 \div 81$

Answer: 342

Exercise $\PageIndex{30}$

$6,510 \div 31$

Exercise $\PageIndex{31}$

$60,536 \div 94$

Answer: 644

Exercise $\PageIndex{32}$

$31,844 \div 38$

Exercise $\PageIndex{33}$

$23,985 \div 45$

Answer: 533

Exercise $\PageIndex{34}$

$60,606 \div 74$

Exercise $\PageIndex{35}$

$2,975,400 \div 285$

Answer: 10,440

Exercise $\PageIndex{36}$

$1,389,660 \div 795$

Exercise $\PageIndex{37}$

$7,162,060 \div 879$

Answer: 8,147 remainder 847

Exercise $\PageIndex{38}$

$7,561,060 \div 909$

Exercise $\PageIndex{39}$

$38 \div 9$

Answer: 4 remainder 2

Exercise $\PageIndex{40}$

$97 \div 4$

Exercise $\PageIndex{41}$

$199 \div 3$

Answer: 66 remainder 1

Exercise $\PageIndex{42}$

$573 \div 6$

Exercise $\PageIndex{43}$

$10,701 \div 13$

Answer: 823 remainder 2

Exercise $\PageIndex{44}$

$13,521 \div 53$

Exercise $\PageIndex{45}$

$3,628 \div 90$

Answer: 40 remainder 28

Exercise $\PageIndex{46}$

$10,592 \div 43$

Exercise $\PageIndex{47}$

$19,965 \div 30$

Answer: 665 remainder 15

Exercise $\PageIndex{48}$

$8,320 \div 21$

Exercise $\PageIndex{49}$

$61,282 \div 64$

Answer: 957 remainder 34

Exercise $\PageIndex{50}$

$1,030 \div 28$

Exercise $\PageIndex{51}$

$7,319 \div 11$

Answer: 665 remainder 4

Exercise $\PageIndex{52}$

$3,628 \div 90$

Exercise $\PageIndex{53}$

$35,279 \div 77$

Answer: 458 remainder 13

Exercise $\PageIndex{54}$

$52,196 \div 68$

Exercise $\PageIndex{55}$

$67,751 \div 68$

Answer: 996 remainder 23

For the following 5 problems, use a calculator to find the quotients.

Exercise $\PageIndex{56}$

$4,346 \div 53$

Exercise $\PageIndex{57}$

$3,234 \div 77$

Answer: 42

Exercise $\PageIndex{58}$

$6,771 \div 37$

Exercise $\PageIndex{59}$

$4,272,320 \div 520$

Answer: 8,216

Exercise $\PageIndex{60}$

$7,558,110 \div 651$

Exercise $\PageIndex{61}$

A mathematics instructor at a high school is paid $17,775 for 9 months. How much money does this instructor make each month?

Answer: $1,975 per month

Exercise $\PageIndex{62}$

A couple pays $4,380 a year for a one-bedroom apartment. How much does this couple pay each month for this apartment?

Exercise $\PageIndex{63}$

Thirty-six people invest a total of $17,460 in a particular stock. If they each invested the same amount, how much did each person invest?

Answer: $485 each person invested

Exercise $\PageIndex{64}$

Each of the 28 students in a mathematics class buys a textbook. If the bookstore sells $644 worth of books, what is the price of each book?

Exercise $\PageIndex{65}$

A certain brand of refrigerator has an automatic ice cube maker that makes 336 ice cubes in one day. If the ice machine makes ice cubes at a constant rate, how many ice cubes does it make each hour?

Answer: 14 cubes per hour

Exercise $\PageIndex{66}$

A beer manufacturer bottles 52,380 ounces of beer each hour. If each bottle contains the same number of ounces of beer, and the manufacturer fills 4,365 bottles per hour, how many ounces of beer does each bottle contain?

Exercise $\PageIndex{67}$

A computer program consists of 68,112 bits. 68,112 bits equals 8,514 bytes. How many bits in one byte?

Answer: 8 bits in each byte

Exercise $\PageIndex{68}$

A 26-story building in San Francisco has a total of 416 offices. If each floor has the same number of offices, how many floors does this building have?

Exercise $\PageIndex{69}$

A college has 67 classrooms and a total of 2,546 desks. How many desks are in each classroom if each classroom has the same number of desks?

Answer: 38

Exercises for Review

Exercise $\PageIndex{70}$

What is the value of 4 in the number 124,621?

Exercise $\PageIndex{71}$

Round 604,092 to the nearest hundred thousand.

Answer: 600,000

Exercise $\PageIndex{72}$

What whole number is the additive identity?

Exercise $\PageIndex{73}$

Find the product. $6,256 \times 100$.

Answer: 625,600

Exercise $\PageIndex{74}$

Find the quotient. $0 \div 11$

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