This document discusses divisibility tests for numbers 3 and 9. It provides examples and explanations of the divisibility rules. Specifically, it states: 1) A number is divisible by 3 if the sum of its digits is divisible by 3. 2) A number is divisible by 9 if the sum of its digits is divisible by 9. 3) Worked examples are provided to demonstrate how to apply these rules to determine if various numbers are divisible by 3 and/or 9.

1. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
Lesson 16 Homework Review
January 13, 2014
Adding:
§
The sum of two even numbers is even.
§
The sum of two odd numbers is even.
§
The sum of an even number and an odd number is odd.
1. 346 + 721
Odd, because the sum of an even and an odd
Mulplying:
§
The product of two even numbers is even.
number is odd.
§
The product of two odd numbers is odd.
§
The product of an even number and an odd number is even.
2. 4,690 x 141
Even, because the product of an even and an odd number is even.
3. 1,462,891 x 745, 629
Odd, because the product of two odd numbers is odd
4. 425, 922 + 32,481,064
Even, because the sum of two even numbers is even.
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2. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
5. 32 + 45 + 67 + 91 + 34 + 56
• The first two addends will be odd, because even and an odd is odd.
• Odd number + 67 will be even because the sum of two odd numbers
is even.
• Even number + 91 will be odd because the sum of an even and an
odd number is odd.
• Odd number + 34 will be odd because the sum of an odd and an even
number is odd.
• Odd number + 56 will be odd because the sum of an odd and an
even number is odd.
Adding:
§
The sum of two even numbers is even.
• Therefore, the final sum will be odd.
§
The sum of two odd numbers is even.
§
The sum of an even number and an odd number is odd.
Mulplying:
§
The product of two even numbers is even.
§
The product of two odd numbers is odd.
§
The product of an even number and an odd number is even.
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3. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
MODULE 2 Arithmec Operaons Including
Division of Fracons
Topic D: Number Theory‐ Thinking Logically
About Mulplicave Arithmec
Lesson 17: Divisibility Tests for 3 and 9
Student Outcomes
§
Students apply divisibility rules, specifically for 3 and 9, to understand factors and mulples.
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4. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
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5. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
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6. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
Discussion
• Divisibility rule for 2: last digit is 0, 2, 4, 6 or 8. (even #)
• Divisibility rule for 4: last two digits are divisible by 4.
• Divisibility rules for 5: last digit is 0 or 5.
• Divisibility rule for 8: last three digits are divisible by 8.
• Divisibility rule for 10: last digit is 0.
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7. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
What do the numbers 12, 18, 30, 66, and 93 all have in common?
Calculate the sum of the digits for each given number. For example, the sum of
the digits in number 12 is 3 because 1 + 2 = 3
12:
18:
30:
66:
93:
What do you notice about all the sums of the digits?
Let's make a claim or theorem....
The divisibility rule for 3: When the sum of the digits is divisible by 3, the
entire number is divisible by 3.
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8. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
What do the numbers 27, 36, 54, 72, and 99 all have in common?
Calculate the sum of the digits for each given number. For example, the
sum of the digits in number 12 is 3 because 1 + 2 = 3
27:
36:
54:
72:
99:
What do you notice about all the sums of the digits?
Let's make a claim or theorem....
The divisibility rule for 9: When the sum of the digits is divisible by 3 and
9, the entire number is divisible by 9.
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9. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
When the sum of the digits is divisible by 3 and 9 , the entire number is divisible
by 9. Let’s try to use this knowledge to determine if a large number is divisible by
3,9 or both.
Is the number 765 is divisible by both 3 and 9?
Find the sum of the digits.
Are 3 and 9 both factors of 18?
Calculating the sum of a number’s digits helps us to determine if the number is
divisible by 3 or 9 or both.
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10. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
§ Divisibility rule for 3: If the sum of the digits is
divisible by 3, then the number is divisible by 3.
§ Divisibility rule for 9: If the sum of the digits is
divisible by 9, then the number is divisible by 9.
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11. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
Example 1
Is 378 divisible by 3 or 9? Why or why not?
a. What are the three digits in the number 378?
b. What is the sum of the three digits?
c. Is 18 divisible by 9?
d. Is the enre number 378 divisible by 9? Why or why not?
The number 378 is divisible by 9 because the sum of the digits is divisible by 9.
e. Is the enre number 378 divisible by 3? Why or why not?
Three is a factor of 378 because if 9 is a factor of 378, then 3 will also be a
factor. OR
The number 378 is divisible by 3 because the sum of the digits is divisible
by 3.
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12. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
Example 2
Is 3,822 divisible by 3 or 9? Why or why not?
Find the sum
Is sum divisible by 3?
Is sum divisible by 9?
Answer:
The number 3,822 is divisible by 3, but not by 9 because the sum of the
digits 3 + 8 + 2 + 2= 15 and 15 is divisible by 3 but not by 9.
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13. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
Divisibility Rules
Divisibility rule for 2: last digit is 0, 2, 4, 6 or 8.
Divisibility rule for 3: if the sum of the digits is divisible by 3
Divisibility rule for 4: last two digits are a number divisible by 4.
Divisibility rules for 5: last digit is 0 or 5.
Divisibility rule for 8: last three digits are a number divisible by 8.
Divisibility rule for 9: If the sum of the digits is divisible by 9
Divisibility rule for 10: last digit is 0.
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14. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
Exercises page s.71
Circle ALL the numbers that are factors of the given number. Complete
any necessary work in the space provided.
1. Is 2,838 divisible by
3
9
4
Divisibility rule for 2: last digit is 0, 2, 4, 6 or 8.
Divisibility rule for 3: if the sum of the digits is divisible by 3
Divisibility rule for 4: last two digits are a number divisible by 4.
Divisibility rules for 5: last digit is 0 or 5.
Divisibility rule for 8: last three digits are a number divisible by 8.
Divisibility rule for 9: If the sum of the digits is divisible by 9
Divisibility rule for 10: last digit is 0.
Explain your reasoning for your choices.
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15. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
2. Is 34,515 divisible by
3
9
January 13, 2014
Divisibility rule for 2: last digit is 0, 2, 4, 6 or 8.
Divisibility rule for 3: if the sum of the digits is divisible by 3
Divisibility rule for 4: last two digits are a number divisible by 4.
Divisibility rules for 5: last digit is 0 or 5.
Divisibility rule for 8: last three digits are a number divisible by 8.
Divisibility rule for 9: If the sum of the digits is divisible by 9
Divisibility rule for 10: last digit is 0.
5
Explain your reasoning for your choices.
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16. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
3. Is 10, 534,341 divisible by
3
9
Divisibility rule for 2: last digit is 0, 2, 4, 6 or 8.
Divisibility rule for 3: if the sum of the digits is divisible by 3
Divisibility rule for 4: last two digits are a number divisible by 4.
Divisibility rules for 5: last digit is 0 or 5.
Divisibility rule for 8: last three digits are a number divisible by 8.
Divisibility rule for 9: If the sum of the digits is divisible by 9
Divisibility rule for 10: last digit is 0.
2
Explain your reasoning for your choices.
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17. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
4. Is 4,320 divisible by
3
January 13, 2014
Divisibility rule for 2: last digit is 0, 2, 4, 6 or 8.
Divisibility rule for 3: if the sum of the digits is divisible by 3
Divisibility rule for 4: last two digits are a number divisible by 4.
Divisibility rules for 5: last digit is 0 or 5.
Divisibility rule for 8: last three digits are a number divisible by 8.
Divisibility rule for 9: If the sum of the digits is divisible by 9
Divisibility rule for 10: last digit is 0.
9
10
Explain your reasoning for your choices.
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18. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
5. Is 6,240 divisible by
3
9
January 13, 2014
Divisibility rule for 2: last digit is 0, 2, 4, 6 or 8.
Divisibility rule for 3: if the sum of the digits is divisible by 3
Divisibility rule for 4: last two digits are a number divisible by 4.
Divisibility rules for 5: last digit is 0 or 5.
Divisibility rule for 8: last three digits are a number divisible by 8.
Divisibility rule for 9: If the sum of the digits is divisible by 9
Divisibility rule for 10: last digit is 0.
8
Explain your reasoning for your choices.
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19. Module 2 Lesson 17 Divisibility Tests for 3 and 9.notebook
January 13, 2014
Closing
Please take out your exit ticket for Lesson 17, close your
binder, and complete the exit ticket. This will be collected.
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