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1.1 Fractions: divisibility rules

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses interesting facts about diving whole numbers. By the end of the module students should be able to recognize a whole number that is divisible by 2, 3, 4, 5, 6, 8, 9, or 10.

Section overview

  • Division by 2, 3, 4, and 5
  • Division by 6, 8, 9, and 10

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 since 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 since 4 + 3 + 2 = 9 size 12{4+5+2=9} {} and 9 is divisible by 3.

432 ÷ 3 = 144 size 12{"432" div 3="144"} {}

The number 25 is not divisible by 3 since 2 + 5 = 7 size 12{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 since the last two digits, 4 and 8, form a number, 48, that is divisible by 4.
31048 ÷ 4 = 7262 size 12{"31048" div 4="7262"} {}
The number 137 is not divisible by 4 since 37 is not divisible by 4.

Division by 5

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

Sample set a

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

Practice set a

State which of the following whole numbers are divisible by 2, 3, 4, or 5. A number may be divisible by more than one number.

26

2

81

3

51

3

385

5

6,112

2, 4

470

2, 5

113,154

2, 3

Division by 6, 8, 9, 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 since its last digit is 4. It is also divisible by 3 since 2 + 3 + 4 = 9 size 12{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 size 12{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 since 000 is divisible by 8.
The number 13,128 is divisible by 8 since 128 is divisible by 8.
The number 1,170 is not divisible by 8 since 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 since 7 + 0 + 2 size 12{7+0+2} {} is divisible by 9.
The number 6588 is divisible by 9 since 6 + 5 + 8 + 8 = 27 size 12{6+5+8+8="27"} {} is divisible by 9.
The number 14,123 is not divisible by 9 since 1 + 4 + 1 + 2 + 3 = 11 size 12{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.

Sample set b

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

Practice set b

State which of the following whole numbers are divisible 6, 8, 9, or 10. Some numbers may be divisible by more than one number.

900

6, 9, 10

6,402

6

6,660

6, 9, 10

55,116

6, 9

Exercises

For the following 30 problems, specify if the whole number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10. Write "none" if the number is not divisible by any digit other than 1. Some numbers may be divisible by more than one number.

48

2, 3, 4, 6, 8

85

30

2, 3, 5, 6, 10

83

98

2

972

892

2, 4

676

903

3

800

223

none

836

665

5

4,381

2,195

5

2,544

5,172

2, 3, 4, 6

1,307

1,050

2, 3, 5, 6, 10

3,898

1,621

none

27,808

45,764

2, 4

49,198

296,122

2

178,656

5,102,417

none

16,990,792

620,157,659

none

457,687,705

Exercises for review

( [link] ) In the number 412, how many tens are there?

1

( [link] ) Subtract 613 from 810.

( [link] ) Add 35, 16, and 7 in two different ways.

( 35 + 16 ) + 7 = 51 + 7 = 58 size 12{ \( "35"+"16" \) +7="51"+7="58"} {}
35 + ( 16 + 7 ) = 35 + 23 = 58 size 12{"35"+ \( "16"+7 \) ="35"+"23"="58"} {}

( [link] ) Find the quotient 35 ÷ 0 size 12{"35" div 0} {} , if it exists.

( [link] ) Find the quotient. 3654 ÷ 42 size 12{"3654" div "42"} {} .

87

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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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