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# Multiplication and Division of Whole Numbers: Multiplication of Whole Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to multiply whole numbers. By the end of the module students should be able to understand the process of multiplication, multiply whole numbers, simplify multiplications with numbers ending in zero, and use a calculator to multiply one whole number by another.

## Section Overview

• Multiplication
• The Multiplication Process With a Single Digit Multiplier
• The Multiplication Process With a Multiple Digit Multiplier
• Multiplication With Numbers Ending in Zero
• Calculators

## Multiplication

Multiplication is a description of repeated addition.

5 + 5 + 5 5 + 5 + 5 size 12{5+5+5} {}

the number 5 is repeated 3 times. Therefore, we say we have three times five and describe it by writing

3 × 5 3 × 5 size 12{3 times 5} {}

Thus,

3 × 5 = 5 + 5 + 5 3 × 5 = 5 + 5 + 5 size 12{3 times 5=5+5+5} {}

### Multiplicand

In a multiplication, the repeated addend (number being added) is called the multi­plicand. In 3×53×5 size 12{3 times 5} {}, the 5 is the multiplicand.

### Multiplier

Also, in a multiplication, the number that records the number of times the multiplicand is used is called the multiplier. In 3×53×5 size 12{3 times 5} {}, the 3 is the multiplier.

### Sample Set A

Express each repeated addition as a multiplication. In each case, specify the multiplier and the multiplicand.

#### Example 1

7 + 7 + 7 + 7 + 7 + 7 7 + 7 + 7 + 7 + 7 + 7 size 12{7+7+7+7+7+7} {}

6×76×7 size 12{6 times 7} {}.


Multiplier is 6.

Multiplicand is 7.

#### Example 2

18 + 18 + 18 18 + 18 + 18 size 12{"18"+"18"+"18"} {}

3×183×18 size 12{3 times "18"} {}.


Multiplier is 3.

Multiplicand is 18.

### Practice Set A

Express each repeated addition as a multiplication. In each case, specify the multiplier and the multiplicand.

#### Exercise 1

12 + 12 + 12 + 12 12 + 12 + 12 + 12 size 12{"12"+"12"+"12"+"12"} {}


. Multiplier is

. Multiplicand is

.

##### Solution

4×124×12 size 12{4 times "12"} {}. Multiplier is 4. Multiplicand is 12.

#### Exercise 2

36 + 36 + 36 + 36 + 36 + 36 + 36 + 36 36 + 36 + 36 + 36 + 36 + 36 + 36 + 36 size 12{"36"+"36"+"36"+"36"+"36"+"36"+"36"+"36"} {}


. Multiplier is

. Multiplicand is

.

##### Solution

8×368×36 size 12{8 times "36"} {}. Multiplier is 8. Multiplicand is 36.

#### Exercise 3

0 + 0 + 0 + 0 + 0 0 + 0 + 0 + 0 + 0 size 12{0+0+0+0+0} {}


. Multiplier is

. Multiplicand is

.

##### Solution

5×05×0 size 12{5 times 0} {}. Multiplier is 5. Multiplicand is 0.

#### Exercise 4

1847 + 1847 + ... + 1847 12,000 times 1847 + 1847 + ... + 1847 12,000 times


. Multiplier is

. Multiplicand is

.

##### Solution

12,000×1,84712,000×1,847 size 12{"12","000" times 1,"847"} {}. Multiplier is 12,000. Multiplicand is 1,847.

### Factors

In a multiplication, the numbers being multiplied are also called factors.

### Products

The result of a multiplication is called the product. In 3×5=153×5=15 size 12{3 times 5="15"} {}, the 3 and 5 are not only called the multiplier and multiplicand, but they are also called factors. The product is 15.

### Indicators of Multiplication ××,⋅,( )

The multiplication symbol (××) is not the only symbol used to indicate multiplication. Other symbols include the dot ( ⋅ ) and pairs of parentheses ( ). The expressions

3×53×5 size 12{3 times 5} {},


3535 size 12{3 cdot 5} {},

3(5)3(5) size 12{3 $$5$$ } {},

(3)5(3)5 size 12{ $$3$$ 5} {},

(3)(5)(3)(5) size 12{ $$3$$ $$5$$ } {}

all represent the same product.

## The Multiplication Process With a Single Digit Multiplier

Since multiplication is repeated addition, we should not be surprised to notice that carrying can occur. Carrying occurs when we find the product of 38 and 7:

First, we compute 7×8=567×8=56 size 12{7 times 8="56"} {}. Write the 6 in the ones column. Carry the 5. Then take 7×3=217×3=21 size 12{7 times 3="21"} {}. Add to 21 the 5 that was carried: 21+5=2621+5=26 size 12{"21"+5="26"} {}. The product is 266.

### Sample Set B

Find the following products.

#### Example 3

3×4=12 Write the 2, carry the 1. 3×6=18 Add to 18 the 1 that was carried:18+1=19. 3×4=12 Write the 2, carry the 1. 3×6=18 Add to 18 the 1 that was carried:18+1=19.

The product is 192.

#### Example 4

5×6=30 Write the 0, carry the 3. 5×2=10 Add to 10 the 3 that was carried:10+3=13. Write the 3, carry the 1. 5×5=25 Add to 25 the 1 that was carried:25+1=6. 5×6=30 Write the 0, carry the 3. 5×2=10 Add to 10 the 3 that was carried:10+3=13. Write the 3, carry the 1. 5×5=25 Add to 25 the 1 that was carried:25+1=6.

The product is 2,630.

#### Example 5

9×4=36 Write the 6, carry the 3. 9×0=0 Add to the 0 the 3 that was carried:0+3=3. Write the 3. 9×8=72 Write the 2, carry the 7. 9×1=9 Add to the 9 the 7 that was carried: 9+7=16. Since there are no more multiplications to perform,write both the 1 and 6. 9×4=36 Write the 6, carry the 3. 9×0=0 Add to the 0 the 3 that was carried:0+3=3. Write the 3. 9×8=72 Write the 2, carry the 7. 9×1=9 Add to the 9 the 7 that was carried: 9+7=16. Since there are no more multiplications to perform,write both the 1 and 6.

The product is 16,236.

### Practice Set B

Find the following products.

#### Exercise 5

37 ×   5 ̲ 37 ×   5 ̲

185

#### Exercise 6

78 ×   8 ̲ 78 ×   8 ̲

624

#### Exercise 7

536 ×     7 ̲ 536 ×     7 ̲

3,752

#### Exercise 8

40,019 ×         8 ̲ 40,019 ×         8 ̲

320,152

#### Exercise 9

301,599 ×           3 ̲ 301,599 ×           3 ̲

904,797

## The Multiplication Process With a Multiple Digit Multiplier

In a multiplication in which the multiplier is composed of two or more digits, the multiplication must take place in parts. The process is as follows:

• Part 1: First Partial Product Multiply the multiplicand by the ones digit of the multiplier. This product is called the first partial product.
• Part 2: Second Partial Product Multiply the multiplicand by the tens digit of the multiplier. This product is called the second partial product. Since the tens digit is used as a factor, the second partial product is written below the first partial product so that its rightmost digit appears in the tens column.
• Part 3: If necessary, continue this way finding partial products. Write each one below the previous one so that the rightmost digit appears in the column directly below the digit that was used as a factor.
• Part 4: Total Product Add the partial products to obtain the total product.

### Note:

It may be necessary to carry when finding each partial product.

### Sample Set C

#### Example 6

Multiply 326 by 48.

• Part 1:

• Part 2:

• Part 3: This step is unnecessary since all of the digits in the multiplier have been used.
• Part 4: Add the partial products to obtain the total product.

• The product is 15,648.

#### Example 7

Multiply 5,369 by 842.

• Part 1:

• Part 2:

• Part 3:

• The product is 4,520,698.

#### Example 8

Multiply 1,508 by 206.

• Part 1:

• Part 2:

Since 0 times 1508 is 0, the partial product will not change the identity of the total product (which is obtained by addition). Go to the next partial product.

• Part 3:

• The product is 310,648

### Practice Set C

#### Exercise 10

Multiply 73 by 14.

1,022

#### Exercise 11

Multiply 86 by 52.

4,472

#### Exercise 12

Multiply 419 by 85.

35,615

#### Exercise 13

Multiply 2,376 by 613.

1,456,488

#### Exercise 14

Multiply 8,107 by 304.

2,464,528

#### Exercise 15

Multiply 66,260 by 1,008.

66,790,080

#### Exercise 16

Multiply 209 by 501.

104,709

#### Exercise 17

Multiply 24 by 10.

240

#### Exercise 18

Multiply 3,809 by 1,000.

3,809,000

#### Exercise 19

Multiply 813 by 10,000.

8,130,000

## Multiplications With Numbers Ending in Zero

Often, when performing a multiplication, one or both of the factors will end in zeros. Such multiplications can be done quickly by aligning the numbers so that the rightmost nonzero digits are in the same column.

### Sample Set D

Perform the multiplication (49,000)(1,200)(49,000)(1,200) size 12{ $$"49","000"$$ $$1,"200"$$ } {}.

(49,000)(1,200) = 49000 ×    1200 ̲ (49,000)(1,200) = 49000 ×    1200 ̲

Since 9 and 2 are the rightmost nonzero digits, put them in the same column.

Draw (perhaps mentally) a vertical line to separate the zeros from the nonzeros.

Multiply the numbers to the left of the vertical line as usual, then attach to the right end of this product the total number of zeros.

The product is 58,800,000

### Practice Set D

#### Exercise 20

Multiply 1,800 by 90.

162,000

#### Exercise 21

Multiply 420,000 by 300.

126,000,000

#### Exercise 22

Multiply 20,500,000 by 140,000.

##### Solution

2,870,000,000,000

## Calculators

Most multiplications are performed using a calculator.

### Sample Set E

#### Example 9

Multiply 75,891 by 263.

 Display Reads Type 75891 75891 Press × 75891 Type 263 263 Press = 19959333

The product is 19,959,333.

#### Example 10

Multiply 4,510,000,000,000 by 1,700.

 Display Reads Type 451 451 Press × 451 Type 17 17 Press = 7667

The display now reads 7667. We'll have to add the zeros ourselves. There are a total of 12 zeros. Attaching 12 zeros to 7667, we get 7,667,000,000,000,000.

The product is 7,667,000,000,000,000.

#### Example 11

Multiply 57,847,298 by 38,976.

 Display Reads Type 57847298 57847298 Press × 57847298 Type 38976 38976 Press = 2.2546563 12

The display now reads 2.2546563 12. What kind of number is this? This is an example of a whole number written in scientific notation. We'll study this concept when we get to decimal numbers.

### Practice Set E

Use a calculator to perform each multiplication.

#### Exercise 23

52×2752×27 size 12{"52" times "27"} {}

1,404

#### Exercise 24

1,448×6,1551,448×6,155 size 12{1,"448" times 6,"155"} {}

8,912,440

#### Exercise 25

8,940,000×205,0008,940,000×205,000 size 12{8,"940","000" times "205","000"} {}

##### Solution

1,832,700,000,000

## Exercises

For the following problems, perform the multiplications. You may check each product with a calculator.

8 × 3 ̲ 8 × 3 ̲

24

3 × 5 ̲ 3 × 5 ̲

8 × 6 ̲ 8 × 6 ̲

48

5 × 7 ̲ 5 × 7 ̲

6 × 16 × 1

6

4 × 54 × 5

75 × 375 × 3

225

35 × 535 × 5

### Exercise 34

45 ×   6 ̲ 45 ×   6 ̲

270

### Exercise 35

31 ×   7 ̲ 31 ×   7 ̲

### Exercise 36

97 ×   6 ̲ 97 ×   6 ̲

582

### Exercise 37

75 × 57 ̲ 75 × 57 ̲

### Exercise 38

64 × 15 ̲ 64 × 15 ̲

960

### Exercise 39

73 × 15 ̲ 73 × 15 ̲

### Exercise 40

81 × 95 ̲ 81 × 95 ̲

7,695

### Exercise 41

31 × 33 ̲ 31 × 33 ̲

57 × 6457 × 64

3,648

76 × 4276 × 42

894 × 52894 × 52

46,488

684 × 38684 × 38

### Exercise 46

115 ×   22 ̲ 115 ×   22 ̲

2,530

### Exercise 47

706 ×   81 ̲ 706 ×   81 ̲

### Exercise 48

328 ×   21 ̲ 328 ×   21 ̲

6,888

### Exercise 49

550 ×   94 ̲ 550 ×   94 ̲

930 × 26930 × 26

24,180

318 × 63318 × 63

### Exercise 52

582 × 127 ̲ 582 × 127 ̲

73,914

### Exercise 53

247 × 116 ̲ 247 × 116 ̲

### Exercise 54

305 × 225 ̲ 305 × 225 ̲

68,625

### Exercise 55

782 × 547 ̲ 782 × 547 ̲

### Exercise 56

771 × 663 ̲ 771 × 663 ̲

511,173

### Exercise 57

638 × 516 ̲ 638 × 516 ̲

### Exercise 58

1,905 × 7101,905 × 710

1,352,550

### Exercise 59

5,757 × 5,0105,757 × 5,010

### Exercise 60

3,106 × 1,752 ̲ 3,106 × 1,752 ̲

5,441,712

### Exercise 61

9,300 × 1,130 ̲ 9,300 × 1,130 ̲

### Exercise 62

7,057 × 5,229 ̲ 7,057 × 5,229 ̲

36,901,053

### Exercise 63

8,051 × 5,580 ̲ 8,051 × 5,580 ̲

### Exercise 64

5,804 × 4,300 ̲ 5,804 × 4,300 ̲

24,957,200

### Exercise 65

357 × 16 ̲ 357 × 16 ̲

### Exercise 66

724 ×     0 ̲ 724 ×     0 ̲

0

### Exercise 67

2,649 ×     41 ̲ 2,649 ×     41 ̲

### Exercise 68

5,173 ×       8 ̲ 5,173 ×       8 ̲

41,384

### Exercise 69

1,999 ×       0 ̲ 1,999 ×       0 ̲

### Exercise 70

1,666 ×       0 ̲ 1,666 ×       0 ̲

0

### Exercise 71

51,730 ×     142 ̲ 51,730 ×     142 ̲

### Exercise 72

387 × 190 ̲ 387 × 190 ̲

73,530

### Exercise 73

3,400 ×     70 ̲ 3,400 ×     70 ̲

### Exercise 74

460,000 ×   14,000 ̲ 460,000 ×   14,000 ̲

6,440,000,000

### Exercise 75

558,000,000 ×         81,000 ̲ 558,000,000 ×         81,000 ̲

### Exercise 76

37,000 ×       120 ̲ 37,000 ×       120 ̲

4,440,000

### Exercise 77

498,000 ×           0 ̲ 498,000 ×           0 ̲

### Exercise 78

4,585,000 ×         140 ̲ 4,585,000 ×         140 ̲

641,900,000

### Exercise 79

30,700,000 ×             180 ̲ 30,700,000 ×             180 ̲

### Exercise 80

8,000 ×      10 ̲ 8,000 ×      10 ̲

80,000

### Exercises for Review

#### Exercise 93

((Reference)) In the number 421,998, how may ten thousands are there?

#### Exercise 94

((Reference)) Round 448,062,187 to the nearest hundred thousand.

448,100,000

#### Exercise 95

((Reference)) Find the sum. 22,451 + 18,976.

#### Exercise 96

((Reference)) Subtract 2,289 from 3,001.

712

#### Exercise 97

((Reference)) Specify which property of addition justifies the fact that (a first whole number + a second whole number) = (the second whole number + the first whole number)

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