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# Dividing Whole Numbers

Summary: This module discussion how to divide whole numbers.

Just like addition and subtraction, multiplication ( × , •) and division ( ÷ , /) are opposites. Multiplying by a number and then dividing by the same number gets us back to the start again:

a × b ÷ b = a

5 × 4 ÷ 4 =5

## Concepts of Division of Whole Numbers

Division is a description of repeated subtraction.

In the process of division, the concern is how many times one number is contained in another number. For example, we might be interested in how many fives are contained in 15. The word times is significant, because it implies a relationship between division and multiplication.

There are several notations used to indicate division. Suppose Q records the number of times 5 is contained in 15. We can indicate this by writing any of the following equations in the following figure:

5Q155Q15 size 12{5 {alignr Q} over {\lline "15"} } {} ─ ─ 5 into 15

155=Q155=Q size 12{ { {"15"} over {5} } =Q} {} ─ ─ 15 divide by 5

15 ÷5 = Q ─ ─ 15 divide by 5

Each of these division notations describes the same number, represented here by the symbol Q. Each notation also converts to the same multiplication form. It is

15 = 5 × 3

In division,

1. The number being divided into is called the dividend.
2. The number dividing into the dividend is called the divisor.
3. The result of the division is called the quotient.

### Examples

Find the following quotients using multiplications facts.

1. Solve for Q: 18 ÷ 6 = Q

• This can be rewritten as: 18 = 6 × Q or 18 = 6 × 3
• Also notice 6 + 6 + 6 = 18
• Therefore, 6 is contained in 18 three times.

2. Solve for P: 243=P243=P size 12{ { {"24"} over {3} } =P} {}

• This can also be rewritten as: 24 = 3 × P or 24 = 3 × 8
• Also note that 8 + 8 + 8 = 24.

3. Solve for R: 6 36 ¯ = R 6 36 ¯ = R size 12{6 {\lline overline "36"} =R} {}

• This can be written as: 36 = 6 × R or 36 = 6 × 6
• There are 6 sixes in 36.

### Exercises

Use multiplication facts to determine the following quotients.

32 ÷ 8

4

18 ÷ 9

2

25 ÷ 5

5

48 ÷ 8

6

28 ÷ 7

4

36 ÷ 4

9

## Division with a Single-Digit Divisor

Our experience with multiplication of whole numbers allows us to perform such divisions as 75 ÷ 5. We perform the division by performing the corresponding multiplication, 5 × Q = 75. Each division we considered in Section 2.2 had a one-digit quotient. Now we will consider divisions in which the quotient may consist of two or more digits. For example, 75 ÷ 5.

Let’s examine the division 75 ÷ 5. We are asked to determine how many 5’s are contained in 75. We approach the problem in the following way.

1. Make an educated guess based on experience with multiplication.
2. Find how close the estimate is by multiplying the estimate by 5.
3. If the product obtained in step 2 is less than 75, find out how much less by subtracting it from 75.
4. 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.

### The Four Steps in Division

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

1. An educated guess
2. A multiplication
3. A subtraction
4. Bringing down the next digit (if necessary)

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

### Single Digit Divisor Example

Find 75 divided by 5.

In summary:

• Divide: 5 goes into 7 one time.
• Multiply: 5 times 1 = 7
• Subtract: 7 minus 5 = 2
• Divide: 5 goes into 25 exactly 5 times

Add: 10 fives plus 5 fives to equal 15.

### Exercises

Perform the following divisions.

28 ÷ 7 =

4

126 ÷ 7 =

18

#### Exercise 9

In the division exercise above, 126 is called the _____________

dividend

#### Exercise 10

In the division exercise above, 7 is called the _____________

divisor

#### Exercise 11

Another name for the answer of a division problem is the ___________

quotient

324 ÷ 4 =

82

520 ÷ 4 =

130

#### Exercise 14

2559325593 size 12{ { {"2559"} over {3} } } {}=

853

#### Exercise 15

636¯636¯ size 12{6 {\lline overline "36"} } {} =

41

#### Exercise 16

5645556455 size 12{ { {"5645"} over {5} } } {} =

1129

#### Exercise 17

8160¯8160¯ size 12{8 {\lline overline "160"} } {} =

20

#### Exercise 18

757,1259757,1259 size 12{ { {"757","125"} over {9} } } {} =

84,125

## Division and Zero

### Division into Zero

Rule: Zero divided by any non-zero whole number is zero.

Let’s look at that happens when the dividend (the number being divided into) is zero, and the divisor (the number doing the dividing) is any whole number except zero. The question is:

What number, if any, is equal to 0X0X size 12{ { {0} over {X} } } {}, where X is any non-zero whole number.

Let’s represent this unknown quotient by Q. Then,

0 X = Q 0 X = Q size 12{ { {0} over {X} } =Q} {}

Converting this division problem to its corresponding multiplication problem, we get Q × X = 0.

From our knowledge of multiplication, we can understand that if the product of two whole numbers is zero, then one or both of the whole numbers must be zero. Since any nonzero whole number (X) is certainly not zero, Q must represent zero. Then, 0X=00X=0 size 12{ { {0} over {X} } =0} {}.

Zero divided by any nonzero whole number is zero.

### Division by Zero

Rule: Division by zero does not name a number. It is, therefore, undefined.

Now we ask, what number, if any is a non-zero whole number divided by zero? Letting Q represent a possible quotient and X representing a non-zero whole number, we get

X 0 = Q X 0 = Q size 12{ { {X} over {0} } =Q} {}
(1)

Converting to the corresponding multiplication form, we have

0 × Q = X

Since Q × 0 = 0, X must be equal to 0. But this is absurd. This would mean that 6 = 0 or 37 = 0. A non-zero whole number cannot equal zero!

Division by zero does not name a number. It is, therefore, undefined.

### Zero Divided by Zero

The division zero into zero is indeterminant.

We are now curious about zero divided by zero 0000 size 12{ { {0} over {0} } } {}. If we let Q represent a potential quotient, we get

0 0 = Q 0 0 = Q size 12{ { {0} over {0} } =Q} {}
(2)

Converting this into multiplication, we get:

Q × 0 = 0

This is a statement that is true regardless of the number used in place of Q. For example,

00=500=25300=500=253alignl { stack { size 12{ { {0} over {0} } =5} {} # size 12{ { {0} over {0} } ="253"} {} } } {} Since 5 times 0 = 0, and 253 times 0 = 0.

A unique quotient cannot be determined. Because the result of the division is inconclusive, we say that it is indeterminant.

The division zero into zero is indeterminant.

### Examples

Perform, if possible, each division.

1. 190190 size 12{ { {"19"} over {0} } } {} This division does not name a whole number, so we say that it is undefined.
2. 140¯140¯ size 12{"14" {\lline overline 0} } {} This is also division by zero and it is undefined.
3. 0808 size 12{ { {0} over {8} } } {} This is equal to zero since diving zero by any whole number results in zero.

### Exercises

What is the result of each of these divisions?

#### Exercise 19

5050 size 12{ { {5} over {0} } } {}=

Undefined

#### Exercise 20

0404 size 12{ { {0} over {4} } } {}=

0

#### Exercise 21

00¯00¯ size 12{0 {\lline overline 0} } {} =

Indeterminant

#### Exercise 22

8080 size 12{ { {8} over {0} } } {}=

Undefined

#### Exercise 23

0707 size 12{ { {0} over {7} } } {}=

0

#### Exercise 24

355355 size 12{ { {"35"} over {5} } } {}=

7

#### Exercise 25

568568 size 12{ { {"56"} over {8} } } {}=

7

## 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.

### Multiple-Digit Divisor Sample

1. Find 2,232 ÷ 36.

2. Find 2,417, 228 divided by 802.

### Exercises

Perform the following divisions.

1376 / 32 =

45

6160 / 55 =

112

18,605 / 61 =

305

144,768 / 48 =

3016

#### Division with a Remainder

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

10 – 4 = 6 – 4 = 2

Because 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.

Because 4 does not divide into 2 (the remainder is less than the divisor) and there are no digits to bring down to continue the process, we are done. We write:

### Exercises

67 / 1 =

67

52 / 4 =

13

776 / 8 =

97

603 / 9 =

67

208 / 4 =

52

21 / 7 =

3

0 / 0 =

0

4264 / 82 =

52

2530 / 55 =

46

55,167 / 71 =

777

#### Exercise 40

##### Solution

$4380 / 12 =$365

#### Exercise 43

A beer manufacturer bottled 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?

##### Solution

52380 / 4365 = 12

#### Exercise 44

A computer program consists of 68,112 bits, which is equal to 8,514 bytes. How many bits are there in one byte?

68112 / 8514 = 8

#### Exercise 45

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?

2546 / 67 = 38

#### Exercise 46

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

336 / 24 = 14

## Some Interesting Facts About Division

Quite often, we are able to determine if a whole number is divisible by another whole number just by observing some simple facts about the number. Some of these facts are listed in this section.

### Division by 2, 3, 4, and 5

#### Division by 2

A whole number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.

The numbers 80, 112, 64, 326, and 1,008 are all divisible by 2 because the last digit of each is 0, 2, 4, 6, or 8, respectively.

The numbers 85 and 731 are not divisible by 2.

#### Division by 3

A whole number is divisible by 3 if the sum of its digits is divisible by 3.

The number 432 is divisible by 3 because 4 + 3 + 2 = 9 and 9 is divisible by 3.

432 ÷ 3 = 144

The number 25 is not divisible by 3 because 2 + 5 =7, and 7 is not divisible by 3.

#### Division by 4

A whole number is divisible by 4 if its last two digits form a number that is divisible by 4.

The number 31,048 is divisible by 4 because the last two digits (4 and 8) form a number (48) that is divisible by 4.

31048 ÷ 4 = 7262

The number 137 is not divisible by 4 because 37 is not divisible by 4.

#### Division by 5

A whole number is divisible by 5 if its last digit is 0 or 5.

The numbers 65, 110, 8,030, and 16,955 are each divisible by 5 because the last digit of each is 0 or 5.

#### Exercises

Which of the following whole numbers are divisible by 2, 3, 4, or 5? A number can be divisible by more than one number.

26

2

81

3

51

3

385

5

6,112

2 and 4

470

2 and 5

113,154

2 and 3

### Division by 6, 8, 9, and 10

#### Division by 6

A number is divisible by 6 if it is divisible by both 2 and 3.

The number 234 is divisible by 2 because its last digit is 4. It is also divisible by 3 because 2 + 3 + 4 = 9 and 9 is divisible by 3. Therefore, 234 is divisible by 6.

The number 6,532 is not divisible by 6. Although its last digit is 2, making it divisible by 2, the sum of its digits, 6 + 5 + 3 + 2 = 16, and 16 is not divisible by 3.

#### Division by 8

A whole number is divisible by 8 if its last three digits form a number that is divisible by 8.

The number 4,000 is divisible by 8 because 000 is divisible by 8.

The number 13,128 is divisible by 8 because 128 is divisible by 8.

The number 1,170 is not divisible by 8 because 170 is not divisible by 8.

#### Division by 9

A whole number is divisible by 9 if the sum of its digits is divisible by 9.

The number 702 is divisible by 9 because 7 + 0 + 2 is divisible by 9.

The number 6,588 is divisible by 9 because 6 + 5 + 8 + 8 = 27 is divisible by 9.

The number 14,123 is not divisible by 9 because 1 + 4 + 1 + 2 + 3 = 11 is not divisible by 9.

#### Division by 10

A whole number is divisible by 10 if its last digit is 0.

The numbers 30, 170, 16,240, and 865,000 are all divisible by 10.

#### Exercises

Which of the following whole numbers are divisible by 6, 8, 9, or 10? Some numbers might be divisible by more than one number.

900

6, 9, and 10

6402

6 only

6,600

6, 8 and 10

##### Exercise 57

55,116

###### Solution

6 and 9

Specify if the number is divisible by 2, 3, 4, 5, 6, 7, 8, or 9? Some numbers may be divisible by more than one number.

##### Exercise 58

48

###### Solution

2, 3, 4, 6, and 8

85

5

30

2, 3, 5, and 6

83

None

98

2 and 7

972

2, 3, 4, 6, 9

892

2 and 4

676

2 and 4

903

3 and 7

800

2, 4, 5, and 8

1,050

2, 3, 5, and 7

4,381

None

2,195

5

##### Exercise 71

2,544

###### Solution

2, 3, 4, 6, and 8

45,764

2 and 4

49,198

2

##### Exercise 74

296,122

###### Solution

2, 3, 4, 6, and 8

176,656

2, 4, and 8

5,102,417

None

620,157,659

None

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