Factors, Divisibility, & Prime / Factors, Divisibility, & Prime / Composite numbersComposite numbers

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Lesson 1d

Factors & DivisibilityFactors & Divisibility

Definition:Definition:A number is A number is divisibledivisible by another number by another number if, when you divide, the remainder is 0.if, when you divide, the remainder is 0.

324824 0

24 is divisibleby 8.

224918 6

24 is not divisible by 9.

Since 24 is divisible by 8, 8 is called a factorfactor of 24.Here are all the factors of 24.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

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2 • 12 = 24,

Factors & DivisibilityFactors & Divisibility cont. . .cont. . .

The number 24 is a The number 24 is a multiplemultiple of each of each of its factors.of its factors.4 • 6 = 24, 3 • 8 = 24,Multiples of 2 are: Multiples of 2 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22,

2424, 26, 28, 30, . . ., 26, 28, 30, . . .

Multiples of 3 are: Multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 3, 6, 9, 12, 15, 18, 21, 2424, 27, 30, , 27, 30, 33, 36, . . . 33, 36, . . .

Multiples of 4 are: Multiples of 4 are: 4, 8, 12, 16, 20, 4, 8, 12, 16, 20, 2424, 28, . . ., 28, . . .

Multiples of 6 are: Multiples of 6 are: 6, 12, 18, 6, 12, 18, 2424, 30, 36, 40, . . ., 30, 36, 40, . . .

Multiples of 8 are:Multiples of 8 are: 8, 16, 8, 16, 2424, 32, 40, 48, 56, . . ., 32, 40, 48, 56, . . .

Multiples of 12 are:Multiples of 12 are: 12, 12, 2424, 36, 48, . . ., 36, 48, . . .

Multiples of 24 are:Multiples of 24 are: 2424, 48, 72, . . ., 48, 72, . . .

1 • 24 = 24

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Factors & DivisibilityFactors & Divisibility cont. . .cont. . .

Thus, Thus, divisible bydivisible by, , factor offactor of, and , and multiple ofmultiple of are related terms. are related terms.

4 • 6 = 24, 12 • 2 = 24,8 • 3 = 24,

24 is 24 is divisible bydivisible by 8. 8.

8 is a 8 is a factor offactor of 24. 24.

24 is a 24 is a multiple ofmultiple of 8. 8.

1 • 24 = 24

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Example 1Example 1

Choose True or False for each Choose True or False for each sentence.sentence.

(a) 12 is a factor of 60. True or False?

(b) 25 is a factor of 60. True or False?

(c) 19 is divisible by 8. True or False?

(d) 30 is divisible by 15. True or False?

(e) 12 is a multiple of 12. True or False?

(f) 36 is a multiple of 9. True or False?

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Answer A is True because :

12 • 5 = 60

Answer B is False because :

260255010

25 is not a factor of 60.

Answer C is False because :

2198162

19 is not divisible by 8.

Answer D is True because :

23015300

30 is divisible by 15.

Answer E is True because :

12 • 1 = 12 12 is a multiple of 12.

Answer F is True because :

9 • 4 = 36 36 is a multiple of 9.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ,13, 14, 15

Example 2Example 2

List all the factors of 15.List all the factors of 15.

① List all the numbers through 15.

Divide 15 by each number in the list. Reject each divisor that produces a nonzero remainder.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ,13, 14, 15

Answer: 1, 3, 5, 15Answer: 1, 3, 5, 15

Your TurnYour Turn

On a piece of paper, list all the On a piece of paper, list all the factors of 9.factors of 9.

Click here to check your answer

Divide 9 by each number in the list. Reject each divisor that produces a nonzero remainder.

1, 2, 3, 4, 5, 6, 7, 8, 9

① List all the numbers through 9.1, 2, 3, 4, 5, 6, 7, 8, 9

Answer: 1, 3, 9Answer: 1, 3, 9

The Process

Your TurnYour Turn

On a piece of paper, list all the On a piece of paper, list all the factors of 22.factors of 22.

Click here to check your answer

Divide 22 by each number in the list. Reject each divisor that produces a nonzero remainder.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 , 15, 16, 17, 18, 19, 20, 21, 22

List all the numbers through 22.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 , 15, 16, 17, 18, 19, 20, 21, 22

Answer: 1, 2, 11, 22Answer: 1, 2, 11, 22

The Process

Your TurnYour Turn

On a piece of paper, list all the On a piece of paper, list all the factors of 5.factors of 5.

Click here to check your answer

1, 2, 3, 4, 5

Divide 5 by each number in the list. Reject each divisor that produces a nonzero remainder.

List all the numbers through 5.

1, 2, 3, 4, 5

Answer: 1, 5Answer: 1, 5

The Process

This is a prime number which we’ll discuss later in the lesson

Divisibility Tests are shortcuts for finding factors of whole numbers.

Here are six useful divisibility tests:

1. A number is divisible by 2 if the last digit is an even number.

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

3. A number is divisible by 4 if the last 2 digits name a multiple of 4.

4. A number is divisible by 5 if the last digit is 0 or 5.

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

6. A number is divisible by 10 if the last digit is 0.Click the text for Examples

Examples:

411 4 + 1 + 1 = 6

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

9,855 9 +8 + 5 + 5 = 27 2 + 7 = 9

Examples:

1036 36 = 4 • 9

A number is divisible by 4 if the last 2 digits name a multiple of 4.

Since 36 is divisible by 4, 1036 is also!

Examples:

10, 25, 40, or 75 are all divisible by 5 because the last digit is a 0 or a 5

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

Examples:

153 1 + 5 + 3 = 9 9 is divisible by itself.

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

468 4 +6 + 8 = 18 9 • 2 = 18

Since 18 is divisible by 9 then 468 is also!

Examples:

10, 20,40, 70, 110, 2190 are all divisible by 10 because the last digit is a 0.

A number is divisible by 10 if the last digit is 0.

PracticeWhich of these numbers are divisible by four?

116 2028 1114Yes Yes No

Click here to review the shortcuts again

Click here to review the shortcuts again

PracticeWhich of these numbers are divisible by three?

51 39 82Yes Yes No

Divisibility Tests are shortcuts for finding factors of whole numbers.

Here are six useful divisibility tests:

1. A number is divisible by 2 if the last digit is an even number.

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

3. A number is divisible by 4 if the last 2 digits name a multiple of 4.

4. A number is divisible by 5 if the last digit is 0 or 5.

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

6. A number is divisible by 10 if the last digit is 0.

DefinitionsPrime NumbersPrime Numbers: A whole number greater than 1 that has only 1 and itself as factors.

Composite NumbersComposite Numbers: A whole number greater than 1 that has at least one factor besides itself and 1.

Prime FactorizationPrime FactorizationPrime Factorization: Every composite number can be expressed as a product of only prime numbers.

Examples:6

2 • 3100

10 • 102 • 5 • 2 • 5

123 • 4

3 • 2 • 2 6 = 2 • 3 12 = 2 • 2 • 3 25 = 2 • 5 • 2 • 5

Choose all of the prime numbers listed above by clicking on them.

You’ll hear chimes and the number will flash if you are correct.

1 2 3 4 5 7 9 116 8 10 1213 15 17 19 21 2314 16 18 20 22 24

Choose, by clicking, all of the composite numbers.

You’ll hear chimes and the number will flash if you are correct.

1 2 3 4 5 6 7 8 109 11 12