Lesson 1 d Factors Divisibility Prime Composite numbers

Lesson 1 d Factors, Divisibility, & Prime / Composite numbers Next

Factors & Divisibility Next Definition: A number is divisible by another number if, when you divide, the remainder is 0. 24 0 24 is divisible by 8. 18 6 24 is not divisible by 9. Since 24 is divisible by 8, 8 is called a factor of 24. Here all the factors of 24. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Factors & Divisibility cont. . . The number 24 is a multiple of each of its factors. 4 • 6 = 24, 2 • 12 = 24, 3 • 8 = 24, 1 • 24 = 24 Multiples of 2 are: Multiples of 3 are: Multiples of 4 are: Multiples of 6 are: Multiples of 8 are: Multiples of 12 are: Multiples of 24 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, . . . 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, . . . 4, 8, 12, 16, 20, 24, 28, . . . 6, 12, 18, 24, 30, 36, 40, . . . 8, 16, 24, 32, 40, 48, 56, . . . 12, 24, 36, 48, . . . 24, 48, 72, . . . Next

Factors & Divisibility cont. . . Thus, divisible by, factor of, and multiple of are related terms. 4 • 6 = 24, 8 • 3 = 24, 12 • 2 = 24, 1 • 24 = 24 24 is divisible by 8. 8 is a factor of 24. 24 is a multiple of 8. Next

Next Example 1 Choose True or False for each 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?

Answer A is True because : 12 • 5 = 60

Answer B is False because : 50 10 25 is not a factor of 60.

Answer C is False because : 16 2 19 is not divisible by 8.

Answer D is True because : 30 0 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.

Example 2 List all the factors of 15. ❶ List all the numbers through 15. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, , 13, 14, 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, 15

Your Turn On a piece of paper, list all the factors of 9. Click here to check your answer

The Process ❶ List all the numbers through 9. 1, 2, 3, 4, 5, 6, 7, 8, 9 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 Answer: 1, 3, 9

Your Turn On a piece of paper, list all the factors of 22. Click here to check your answer

The Process 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 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 Answer: 1, 2, 11, 22

Your Turn On a piece of paper, list all the factors of 5. Click here to check your answer

The Process List all the numbers through 5. 1, 2, 3, 4, 5 Divide 5 by each number in the list. Reject each divisor that produces a nonzero remainder. Answer: 1, 5 1, 2, 3, 4, 5 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

A number is divisible by 3 if the sum of its digits is divisible by 3. Examples: 411 4 + 1 = 6 9, 855 9 +8 + 5 = 27 2 + 7 = 9

A number is divisible by 4 if the last 2 digits name a multiple of 4. Examples: 1036 36 = 4 • 9 Since 36 is divisible by 4, 1036 is also!

A number is divisible by 5 if the last digit is 0 or 5. 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 9 if the sum of its digits is divisible by 9. Examples: 153 1 + 5 + 3 = 9 9 is divisible by itself. 468 4 +6 + 8 = 18 9 • 2 = 18 Since 18 is divisible by 9 then 468 is also!

A number is divisible by 10 if the last digit is 0. Examples: 10, 20, 40, 70, 110, 2190 are all divisible by 10 because the last digit is a 0.

Practice Which of these numbers are divisible by four? 116 Yes 2028 Yes Click here to review the shortcuts again 1114 No

Practice Which of these numbers are divisible by three? 51 Yes 39 Yes Click here to review the shortcuts again 82 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.

Definitions Prime Numbers: Numbers A whole number greater than 1 that has only 1 and itself as factors. Composite Numbers: Numbers A whole number greater than 1 that has at least one factor besides itself and 1.

Prime Factorization: Factorization Every composite number can be expressed as a product of only prime numbers. 6 2 • 3 6=2 • 3 Examples: 12 3 • 4 3 • 2 100 10 • 10 2 • 5 • 2 • 5 12 = 2 • 3 25 = 2 • 5 • 2 • 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 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 6 7 8 9 10 11 12 Choose, by clicking, all of the composite numbers. You’ll hear chimes and the number will flash if you are correct.