1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 1.8 Tests for Divisibility (2, 3, 4, 5, 6, 9, and 10)

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Know the tests for checking divisibility by 2, 3, 4, 5, 6, 9, and 10. o Apply the concept of divisibility to products of whole numbers.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Tests for Divisibility by 2, 3, 4, 5, 6, 9, and 10 Divisibility If a number can be divided by another number so that the remainder is 0, then we say: The dividend is exactly divisible by (or is divisible by) the divisor. Or, the divisor divides the dividend. For example, 46 ÷ 2 = 23, so 46 is divisible by 2. Or, 2 divides 46.

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Even and Odd Whole Numbers Even whole numbers are divisible by 2. (If a whole number is divided by 2 and the remainder is 0, then the whole number is even.) Odd whole numbers are not divisible by 2. (If a whole number is divided by 2 and the remainder is 1, then the whole number is odd.) (Note: Every whole number is either even or odd.) Tests for Divisibility by 2, 3, 4, 5, 6, 9, and 10

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Tests for Divisibility by 2, 3, 4, 5, 6, 9, and 10 Divisibility by 2 If the last digit (ones digit) of a whole number is 0, 2, 4, 6, or 8 (an even digit), then the number is divisible by 2 (the number is even).

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Divisibility by 2 Determine which of the following numbers are divisible by 2. a.674 Solution 674 is divisible by 2 because the ones digit is 4 (an even digit).

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Divisibility by 2 (cont.) b. 357 Solution 357 is not divisible by 2 because the ones digit is not 0, 2, 4, 6, or 8.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Tests for Divisibility by 2, 3, 4, 5, 6, 9, and 10 Divisibility by 3 If the sum of the digits of a whole number is divisible by 3, then the number is divisible by 3.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Divisibility by 3 Determine which of the following numbers are divisible by 3. a.6801 Solution 6801 is divisible by 3 because 6 + 8 + 0 + 1 = 15, and 15 is divisible by 3.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Divisibility by 3 (cont.) b. 356 Solution 356 is not divisible by 3 because 3 + 5 + 6 = 14, and 14 is not divisible by 3.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Tests for Divisibility by 2, 3, 4, 5, 6, 9, and 10 Divisibility by 4 If the last two digits of a whole number form a number that is divisible by 4, then the number is divisible by 4. (00 is considered to be divisible by 4.)

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Divisibility by 4 Determine which of the following numbers are divisible by 4. a.9036 Solution 9036 is divisible by 4 because 36 (the number formed by the last two digits) is divisible by 4.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Divisibility by 4 (cont.) b. 6700 Solution 6700 is divisible by 4 because 00 is considered to be divisible by 4.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Divisibility by 4 (cont.) c. 15,031 Solution 15,031 is not divisible by 4 because 31 is not divisible by 4.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Tests for Divisibility by 2, 3, 4, 5, 6, 9, and 10 Divisibility by 5 If the last digit (ones digit) of a whole number is 0 or 5, then the number is divisible by 5.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Divisibility by 5 Determine which of the following numbers are divisible by 5. a.1365b. 970c. 1863 Solution a.1365 is divisible by 5 because the ones digit is 5. b.970 is divisible by 5 because the ones digit is 0. c.1863 is not divisible by 5 because the ones digit is not 0 or 5.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Tests for Divisibility by 2, 3, 4, 5, 6, 9, and 10 Divisibility by 6 If a whole number is divisible by both 2 and 3, then the number is divisible by 6.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Divisibility by 6 Determine which of the following numbers are divisible by 6. a.9054 Solution 9054 is divisible by 2 because the ones digit is 4. 9054 is divisible by 3 because 9 + 0 + 5 + 4 = 18, and 18 is divisible by 3. Therefore, 9054 is divisible by 6.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Divisibility by 6 (cont.) b. 17,000 Solution 17,000 is divisible by 2 because the ones digit is 0. 17,000 is not divisible by 3 because 1 + 7 + 0 + 0 + 0 = 8, and 8 is not divisible by 3. Therefore, 17,000 is not divisible by 6.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Tests for Divisibility by 2, 3, 4, 5, 6, 9, and 10 Divisibility by 9 If the sum of the digits of a whole number is divisible by 9, then the number is divisible by 9.

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Divisibility by 9 Determine which of the following numbers are divisible by 9. a.2530 Solution 2530 is not divisible by 9 because 2 + 5 + 3 + 0 = 10, and 10 is not divisible by 9.

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Divisibility by 9 (cont.) b. 873 Solution 873 is divisible by 9 because 8 + 7 + 3 = 18, and 18 is divisible by 9.

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Tests for Divisibility by 2, 3, 4, 5, 6, 9, and 10 Divisibility by 10 If the last digit (ones digit) of a whole number is 0, then the number is divisible by 10.

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Divisibility by 10 Determine which of the following numbers are divisible by 10. a.12,530 Solution 12,530 is divisible by 10 because the ones digit is 0.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Divisibility by 10 (cont.) b. 841 Solution 841 is not divisible by 10 because the ones digit is not 0.

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 8: Divisibility Rules a.250 is divisible by 10 because ________________________ b.5712 is divisible by 4 because ________________________ c.5402 is not divisible by 3 because ________________________ d.6036 is divisible by 6 because ________________________ the last digit is 0. the number formed by the last 2 digits (12) is divisible by 4. 5 + 4 + 0 + 2 = 11, and 11 is not divisible by 3. 6036 is divisible by both 2 and 3.

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 9: Divisibility of Products Does 36 divide the product 3 ⋅ 4 ⋅ 5 ⋅ 7 ⋅ 9? If so, how many times? Solution Because 36 = 4 ⋅ 9, we have Thus 36 divides the product 105 times. = (4 ⋅ 9) (3 ⋅ 5 ⋅ 7)3 ⋅ 4 ⋅ 5 ⋅ 7 ⋅ 9 = 36 ⋅ 105

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Divisibility of Products Does 15 divide the product 5 ⋅ 7 ⋅ 2 ⋅ 3 ⋅ 2? If so, how many times? Solution Because 15 = 3 ⋅ 5, we have Thus 15 divides the product 28 times. = 15 ⋅ 28 5 ⋅ 7 ⋅ 2 ⋅ 3 ⋅ 2= (3 ⋅ 5)(7 ⋅ 2 ⋅ 2)

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Divisibility of Products Does 35 divide the product 3 ⋅ 4 ⋅ 5 ⋅ 11? If so, how many times? Solution We know that 35 = 5 ⋅ 7 and even though 5 is a factor of the product, 7 is not. Therefore, 35 does not divide the product 3 ⋅ 4 ⋅ 5 ⋅ 11. In other words, 3 ⋅ 4 ⋅ 5 ⋅ 11 = 660; 660 is not divisible by 35.

30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 12: Divisibility of Products Does 77 divide the product 3 ⋅ 11 ⋅ 6 ⋅ 7 ⋅ 2? If so, how many times? Solution Because 77 = ____ ⋅ ____, we have 3 ⋅ 11 ⋅ 6 ⋅ 7 ⋅ 2 = ( ___ ⋅ ___ )( ___ ⋅ ___ ⋅ ___ ) =( ___ )( ___ ) Thus 77 divides the product _____ times. 36 77 32117 7 6

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems 1.Using the techniques of this section, determine which of the numbers 2, 3, 4, 5, 6, 9, and 10 (if any) will divide into each of the following numbers. a. 842b. 675c. 9030d. 4031 2.Does 16 divide the product 3 ⋅ 5 ⋅ 4 ⋅ 7 ⋅ 4? If so, how many times?

32. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.a.2 b.3, 5, 9 c.2, 3, 5, 6, 10 d.none 2.yes, 105 times