1. Slide 1 Lesson 35 Testing for Divisibility WO.17Use long division to determine if one number is divisible by another. WO.23Use divisibility rules to determine if a number is divisible by 2, 3, 5, or 9 and understand the justification for these rules. Chapter 7 Lesson 35

2. Slide 2 Lesson 35 Objectives Understand and use the divisibility rules for 2, 3, 5 and 9.

3. Slide 3 Lesson 35 Remember from Before What is a factor? What is a multiple? How are multiples and factors related?

4. Slide 4 Lesson 35 Get Your Brain in Gear 1. Use mental math to divide 369 by 9. 2. Use mental math to divide 85 by 5. 41 17

5. Slide 5 Lesson 35 Multiples of 2. All multiples of 2 can be expressed as the repeated addition of 2. 10 = 2 + 2 + 2 + 2 + 2

6. Slide 6 Lesson 35 Is 36 divisible by 2? Let’s try to express 36 as repeated addition of 2.

7. Slide 7 Lesson 35 Let’s try 21. We have a unit square left over. This means that 21 is not divisible by 2.

8. Slide 8 Lesson 35 What about larger powers of 10? 100 = 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 100 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 Since all the powers of ten are multiples of 10, they also are all multiples of 2.

9. Slide 9 Lesson 35 Divisible by 2 rule: If a whole number ends in 0, 2, 4, 6 or 8, then the number is divisible by 2. Otherwise it is not divisible by 2.

10. Slide 10 Lesson 35 Applying the rule, is the following number divisible by 2? 47,297,593 The digit in the 10 0 place is 3, and 3 is not divisible by 2.

11. Slide 11 Lesson 35 Check for Understanding 1. Determine whether the number is divisible by 2. a. 23 b. 78 c. 504 d. 8,241 e. 6,794 Not divisible by 2. Divisible by 2. Not divisible by 2. Divisible by 2.

12. Slide 12 Lesson 35 Divisibility by 5 Is 10 divisible by 5? 10 = 5 + 5 Since 10 is divisible by 5, so are all the larger powers of 10.

13. Slide 13 Lesson 35 Divisibility by 5 rule: If a whole number ends in 0 or 5, then the number is divisible by 5. Otherwise it is not divisible by 5. According to this rule, would 365 be divisible by 5?

14. Slide 14 Lesson 35 Check for Understanding 2. Determine if the number is divisible by 5. a. 70 b. 553 d. 72865 c. 10003 e. 8003000 Divisible by 5. Not divisible by 5.

15. Slide 15 Lesson 35 Divisibility by 9 Since 10 is not divisible by 9, we can’t simply check the last digit. Let’s see if 27 is divisible by 9: 27 = 9 + 9 + 9

16. Slide 16 Lesson 35 When testing for divisibility by 9, we see that each 10 leaves 1 left over, so we can treat each 10 as a 1. Is 52 divisible by 9? Since 5 + 2 equals 7, we conclude 52 is not divisible by 9.

17. Slide 17 Lesson 35 Is 63 divisible by 9? Remember, each 10 is treated as a 1. Since 6 + 3 equals 9, this means 63 is divisible by 9. Is 85 divisible by 9? How do you know?

18. Slide 18 Lesson 35 7 + 5 + 6 = 18 Since 18 is divisible by 9, we conclude that 756 is also divisible by 9. What about larger numbers? Is 756 divisible by 9?

19. Slide 19 Lesson 35 If the digits of a whole number add up to a multiple of 9, then the number is divisible by 9. Otherwise it is not divisible by 9.

20. Slide 20 Lesson 35 Check for Understanding 3. Determine whether the number is divisible by 9. a. 73 b. 108 c. 7812 d. 6873 e. 98016 Not divisible by 9. Divisible by 9.

21. Slide 21 Lesson 35 Let’s develop a test for divisibility by 3. Let’s check if 42 is divisible by 3.

22. Slide 22 Lesson 35 If the digits of a whole number add up to a multiple of 3, then the number is divisible by 3. Otherwise it is not divisible by 3. Is 592 divisible by 3? 5 + 9 + 2 = 16 1 + 6 = 7

23. Slide 23 Lesson 35 We can verify that 592 is not divisible by 3 using long division:

24. Slide 24 Lesson 35 Check for Understanding 4. Test whether the number is divisible by 3. Verify the result using long division. 5. Using what you learned in this lesson, how can you quickly determine if 1,335 is divisible by 15? Is it? 6. What is the smallest number you can add to 7,120 to make it divisible by 3? 7. When you divide 2,349,684 by 5, will there be a remainder? What will the remainder be? a. 84b. 275c. 1086 d. 23938 e. 62505 Yes No You check to see if it is divisible by 3 and divisible by 5. Thus 1,335 is divisible by 15. Add 2. 7,122 is divisible by 3. Yes; 4

25. Slide 25 Lesson 35 Multiple Choice Practice 1. Which of the following numbers is NOT a factor of 29,910? 2 3 5 9

26. Slide 26 Lesson 35 A student made the following claims about divisibility. What is the student misunderstanding? What would you tell this student to correct their understanding? Find the Errors The student was able to correctly determine if a number is divisible by 2 or 5, but misunderstood how to test for divisibility of 3. You cannot in general look at the last digit to determine if it is divisible by 3, you must add all the digits together and check if the number is a multiple of 3. 5 + 2 + 3 = 10, which is not a multiple of 3. Therefore, 523 is not divisible by 3.