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# 1 f6 some facts about the disvisibility of numbers

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### 1 f6 some facts about the disvisibility of numbers

1. 1. We start out with a simple mathematics procedure that is often used in real live. Some Facts About Divisibility
2. 2. We start out with a simple mathematics procedure that is often used in real live. It’s called the digit sum. Some Facts About Divisibility
3. 3. We start out with a simple mathematics procedure that is often used in real live. It’s called the digit sum. Just as its name suggests, we sum all the digits in a number. Some Facts About Divisibility
4. 4. We start out with a simple mathematics procedure that is often used in real live. It’s called the digit sum. Just as its name suggests, we sum all the digits in a number. The digit sum of 12 is 1 + 2 = 3, Some Facts About Divisibility
5. 5. We start out with a simple mathematics procedure that is often used in real live. It’s called the digit sum. Just as its name suggests, we sum all the digits in a number. The digit sum of 12 is 1 + 2 = 3, is the same as the digit sum of 21, 111, or 11100. Some Facts About Divisibility
6. 6. Some Facts About Divisibility
7. 7. We start out with a simple mathematics procedure that is often used in real live. It’s called the digit sum. Just as its name suggests, we sum all the digits in a number. The digit sum of 12 is 1 + 2 = 3, is the same as the digit sum of 21, 111, or 11100. To find the digit sum of Some Facts About Divisibility 7 8 9 1 8 2 7 3
8. 8. We start out with a simple mathematics procedure that is often used in real live. It’s called the digit sum. Just as its name suggests, we sum all the digits in a number. The digit sum of 12 is 1 + 2 = 3, is the same as the digit sum of 21, 111, or 11100. To find the digit sum of Some Facts About Divisibility add 7 8 9 1 8 2 7 3 15
9. 9. We start out with a simple mathematics procedure that is often used in real live. It’s called the digit sum. Just as its name suggests, we sum all the digits in a number. The digit sum of 12 is 1 + 2 = 3, is the same as the digit sum of 21, 111, or 11100. To find the digit sum of Some Facts About Divisibility add 7 8 9 1 8 2 7 3 15 30
10. 10. We start out with a simple mathematics procedure that is often used in real live. It’s called the digit sum. Just as its name suggests, we sum all the digits in a number. The digit sum of 12 is 1 + 2 = 3, is the same as the digit sum of 21, 111, or 11100. To find the digit sum of Some Facts About Divisibility add 7 8 9 1 8 2 7 3 15 30 add 45
11. 11. We start out with a simple mathematics procedure that is often used in real live. It’s called the digit sum. Just as its name suggests, we sum all the digits in a number. The digit sum of 12 is 1 + 2 = 3, is the same as the digit sum of 21, 111, or 11100. To find the digit sum of Some Facts About Divisibility add 7 8 9 1 8 2 7 3 15 30 add 45 Hence the digit sum of 78999111 is 45.
12. 12. We start out with a simple mathematics procedure that is often used in real live. It’s called the digit sum. Just as its name suggests, we sum all the digits in a number. The digit sum of 12 is 1 + 2 = 3, is the same as the digit sum of 21, 111, or 11100. To find the digit sum of Some Facts About Divisibility add 7 8 9 1 8 2 7 3 15 30 add 45 Hence the digit sum of 78999111 is 45. If we keep adding the digits, the sums eventually become a single digit sum – the digit root. 9
13. 13. We start out with a simple mathematics procedure that is often used in real live. It’s called the digit sum. Just as its name suggests, we sum all the digits in a number. The digit sum of 12 is 1 + 2 = 3, is the same as the digit sum of 21, 111, or 11100. To find the digit sum of Some Facts About Divisibility add 7 8 9 1 8 2 7 3 15 30 add 45 Hence the digit sum of 78999111 is 45. If we keep adding the digits, the sums eventually become a single digit sum – the digit root. The digit root of 78198273 is 9. 9
14. 14. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? Some Facts About Divisibility
15. 15. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? In mathematics, we ask “is 384 divisible by 12?” for short. Some Facts About Divisibility
16. 16. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? In mathematics, we ask “is 384 divisible by 12?” for short. How about 2,349,876,543,214 pieces of chocolate with 18 kids? Some Facts About Divisibility
17. 17. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? In mathematics, we ask “is 384 divisible by 12?” for short. How about 2,349,876,543,214 pieces of chocolate with 18 kids? Is 2,349,876,543,214 divisible by 18? Some Facts About Divisibility
18. 18. One simple application of the digit sum is to check if a number may be divided completely by numbers such as 9, 12, or 18. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? In mathematics, we ask “is 384 divisible by 12?” for short. How about 2,349,876,543,214 pieces of chocolate with 18 kids? Is 2,349,876,543,214 divisible by 18? Some Facts About Divisibility
19. 19. One simple application of the digit sum is to check if a number may be divided completely by numbers such as 9, 12, or 18. I. The Digit Sum Test for Divisibility by 3 and 9. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? In mathematics, we ask “is 384 divisible by 12?” for short. How about 2,349,876,543,214 pieces of chocolate with 18 kids? Is 2,349,876,543,214 divisible by 18? Some Facts About Divisibility
20. 20. One simple application of the digit sum is to check if a number may be divided completely by numbers such as 9, 12, or 18. I. The Digit Sum Test for Divisibility by 3 and 9. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? In mathematics, we ask “is 384 divisible by 12?” for short. How about 2,349,876,543,214 pieces of chocolate with 18 kids? Is 2,349,876,543,214 divisible by 18? If the digit sum or digit root of a number may be divided by 3 (or 9) then the number itself maybe divided by 3 (or 9). Some Facts About Divisibility
21. 21. One simple application of the digit sum is to check if a number may be divided completely by numbers such as 9, 12, or 18. I. The Digit Sum Test for Divisibility by 3 and 9. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? In mathematics, we ask “is 384 divisible by 12?” for short. How about 2,349,876,543,214 pieces of chocolate with 18 kids? Is 2,349,876,543,214 divisible by 18? If the digit sum or digit root of a number may be divided by 3 (or 9) then the number itself maybe divided by 3 (or 9). For example 12, 111, 101010 and 300100200111 all have digit sums that may be divided by 3, Some Facts About Divisibility
22. 22. One simple application of the digit sum is to check if a number may be divided completely by numbers such as 9, 12, or 18. I. The Digit Sum Test for Divisibility by 3 and 9. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? In mathematics, we ask “is 384 divisible by 12?” for short. How about 2,349,876,543,214 pieces of chocolate with 18 kids? Is 2,349,876,543,214 divisible by 18? If the digit sum or digit root of a number may be divided by 3 (or 9) then the number itself maybe divided by 3 (or 9). For example 12, 111, 101010 and 300100200111 all have digit sums that may be divided by 3, therefore all of them may be divided by 3 evenly. Some Facts About Divisibility
23. 23. One simple application of the digit sum is to check if a number may be divided completely by numbers such as 9, 12, or 18. I. The Digit Sum Test for Divisibility by 3 and 9. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? In mathematics, we ask “is 384 divisible by 12?” for short. How about 2,349,876,543,214 pieces of chocolate with 18 kids? Is 2,349,876,543,214 divisible by 18? If the digit sum or digit root of a number may be divided by 3 (or 9) then the number itself maybe divided by 3 (or 9). For example 12, 111, 101010 and 300100200111 all have digit sums that may be divided by 3, therefore all of them may be divided by 3 evenly. However only 3001002000111, whose digit sum is 9, Some Facts About Divisibility
24. 24. One simple application of the digit sum is to check if a number may be divided completely by numbers such as 9, 12, or 18. I. The Digit Sum Test for Divisibility by 3 and 9. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? In mathematics, we ask “is 384 divisible by 12?” for short. How about 2,349,876,543,214 pieces of chocolate with 18 kids? Is 2,349,876,543,214 divisible by 18? If the digit sum or digit root of a number may be divided by 3 (or 9) then the number itself maybe divided by 3 (or 9). For example 12, 111, 101010 and 300100200111 all have digit sums that may be divided by 3, therefore all of them may be divided by 3 evenly. However only 3001002000111, whose digit sum is 9, may be divided evenly by 9. Some Facts About Divisibility
25. 25. One simple application of the digit sum is to check if a number may be divided completely by numbers such as 9, 12, or 18. I. The Digit Sum Test for Divisibility by 3 and 9. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? In mathematics, we ask “is 384 divisible by 12?” for short. How about 2,349,876,543,214 pieces of chocolate with 18 kids? Is 2,349,876,543,214 divisible by 18? If the digit sum or digit root of a number may be divided by 3 (or 9) then the number itself maybe divided by 3 (or 9). For example 12, 111, 101010 and 300100200111 all have digit sums that may be divided by 3, therefore all of them may be divided by 3 evenly. However only 3001002000111, whose digit sum is 9, may be divided evenly by 9. Example A. Identify which of the following numbers are divisible by 3 and which are divisible by 9 by inspection. a. 2345 b. 356004 c. 6312 d. 870480 Some Facts About Divisibility
26. 26. We refer the above digit–sum check for 3 and 9 as test I. Some Facts About Divisibility
27. 27. We refer the above digit–sum check for 3 and 9 as test I. We continue with test II and III. Some Facts About Divisibility
28. 28. II. The Test for Divisibility by 2, 4, and 8 We refer the above digit–sum check for 3 and 9 as test I. We continue with test II and III. Some Facts About Divisibility
29. 29. II. The Test for Divisibility by 2, 4, and 8 A number is divisible by 2 if its last digit is even. We refer the above digit–sum check for 3 and 9 as test I. We continue with test II and III. Some Facts About Divisibility
30. 30. II. The Test for Divisibility by 2, 4, and 8 A number is divisible by 2 if its last digit is even. We refer the above digit–sum check for 3 and 9 as test I. A number is divisible by 4 if its last 2 digits is divisible by 4 – you may ignore all the digits in front of them. We continue with test II and III. Some Facts About Divisibility
31. 31. II. The Test for Divisibility by 2, 4, and 8 A number is divisible by 2 if its last digit is even. We refer the above digit–sum check for 3 and 9 as test I. Hence ****32 is divisible by 4 A number is divisible by 4 if its last 2 digits is divisible by 4 – you may ignore all the digits in front of them. We continue with test II and III. Some Facts About Divisibility
32. 32. II. The Test for Divisibility by 2, 4, and 8 A number is divisible by 2 if its last digit is even. We refer the above digit–sum check for 3 and 9 as test I. Hence ****32 is divisible by 4 but ****42 is not. A number is divisible by 4 if its last 2 digits is divisible by 4 – you may ignore all the digits in front of them. We continue with test II and III. Some Facts About Divisibility
33. 33. II. The Test for Divisibility by 2, 4, and 8 A number is divisible by 2 if its last digit is even. We refer the above digit–sum check for 3 and 9 as test I. Hence ****32 is divisible by 4 but ****42 is not. A number is divisible by 8 if its last 3 digits is divisible by 8. A number is divisible by 4 if its last 2 digits is divisible by 4 – you may ignore all the digits in front of them. We continue with test II and III. Some Facts About Divisibility
34. 34. II. The Test for Divisibility by 2, 4, and 8 A number is divisible by 2 if its last digit is even. We refer the above digit–sum check for 3 and 9 as test I. Hence ****32 is divisible by 4 but ****42 is not. A number is divisible by 8 if its last 3 digits is divisible by 8. Hence ****880 is divisible by 8, A number is divisible by 4 if its last 2 digits is divisible by 4 – you may ignore all the digits in front of them. We continue with test II and III. Some Facts About Divisibility
35. 35. II. The Test for Divisibility by 2, 4, and 8 A number is divisible by 2 if its last digit is even. We refer the above digit–sum check for 3 and 9 as test I. Hence ****32 is divisible by 4 but ****42 is not. A number is divisible by 8 if its last 3 digits is divisible by 8. Hence ****880 is divisible by 8, but ****820 is not. A number is divisible by 4 if its last 2 digits is divisible by 4 – you may ignore all the digits in front of them. We continue with test II and III. Some Facts About Divisibility
36. 36. II. The Test for Divisibility by 2, 4, and 8 A number is divisible by 2 if its last digit is even. We refer the above digit–sum check for 3 and 9 as test I. Hence ****32 is divisible by 4 but ****42 is not. A number is divisible by 8 if its last 3 digits is divisible by 8. Hence ****880 is divisible by 8, but ****820 is not. A number is divisible by 4 if its last 2 digits is divisible by 4 – you may ignore all the digits in front of them. III. The Test for Divisibility by 5 We continue with test II and III. Some Facts About Divisibility
37. 37. II. The Test for Divisibility by 2, 4, and 8 A number is divisible by 2 if its last digit is even. We refer the above digit–sum check for 3 and 9 as test I. Hence ****32 is divisible by 4 but ****42 is not. A number is divisible by 8 if its last 3 digits is divisible by 8. Hence ****880 is divisible by 8, but ****820 is not. A number is divisible by 4 if its last 2 digits is divisible by 4 – you may ignore all the digits in front of them. III. The Test for Divisibility by 5. A number is divisible by 5 if its last digit is 5 or 0. We continue with test II and III. Some Facts About Divisibility
38. 38. II. The Test for Divisibility by 2, 4, and 8 A number is divisible by 2 if its last digit is even. We refer the above digit–sum check for 3 and 9 as test I. Hence ****32 is divisible by 4 but ****42 is not. A number is divisible by 8 if its last 3 digits is divisible by 8. Hence ****880 is divisible by 8, but ****820 is not. A number is divisible by 4 if its last 2 digits is divisible by 4 – you may ignore all the digits in front of them. III. The Test for Divisibility by 5. A number is divisible by 5 if its last digit is 5 or 0. From the above checks, we get the following checks for important numbers such as 6, 12, 15, 18, 36, etc.. We continue with test II and III. Some Facts About Divisibility
39. 39. II. The Test for Divisibility by 2, 4, and 8 A number is divisible by 2 if its last digit is even. We refer the above digit–sum check for 3 and 9 as test I. Hence ****32 is divisible by 4 but ****42 is not. A number is divisible by 8 if its last 3 digits is divisible by 8. Hence ****880 is divisible by 8, but ****820 is not. A number is divisible by 4 if its last 2 digits is divisible by 4 – you may ignore all the digits in front of them. III. The Test for Divisibility by 5. A number is divisible by 5 if its last digit is 5 or 0. From the above checks, we get the following checks for important numbers such as 6, 12, 15, 18, 36, etc.. The idea is to do multiple checks on any given numbers. We continue with test II and III. Some Facts About Divisibility
40. 40. The Multiple Checks Principle Some Facts About Divisibility
41. 41. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. Some Facts About Divisibility
42. 42. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. The number 102 is divisible by 3-via the digit sum test. Some Facts About Divisibility
43. 43. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. The number 102 is divisible by 3-via the digit sum test. It is divisible by 2 because the last digit is even. Some Facts About Divisibility
44. 44. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. The number 102 is divisible by 3-via the digit sum test. It is divisible by 2 because the last digit is even. Hence 102 is divisible by 2*3 = 6. Some Facts About Divisibility
45. 45. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. The number 102 is divisible by 3-via the digit sum test. It is divisible by 2 because the last digit is even. Hence 102 is divisible by 2*3 = 6 (102 = 6 *17). Some Facts About Divisibility
46. 46. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. The number 102 is divisible by 3-via the digit sum test. It is divisible by 2 because the last digit is even. Hence 102 is divisible by 2*3 = 6 (102 = 6 *17). Example B. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? How about 2,349,876,543,214 pieces of chocolate with 18 kids? Some Facts About Divisibility
47. 47. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. The number 102 is divisible by 3-via the digit sum test. It is divisible by 2 because the last digit is even. Hence 102 is divisible by 2*3 = 6 (102 = 6 *17). Example B. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? How about 2,349,876,543,214 pieces of chocolate with 18 kids? For 384 its digit sum is 15 so it’s divisible by 3 (but not 9). Some Facts About Divisibility
48. 48. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. The number 102 is divisible by 3-via the digit sum test. It is divisible by 2 because the last digit is even. Hence 102 is divisible by 2*3 = 6 (102 = 6 *17). Example B. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? How about 2,349,876,543,214 pieces of chocolate with 18 kids? For 384 its digit sum is 15 so it’s divisible by 3 (but not 9). Its last two digits are 84 which is divisible by 4. Some Facts About Divisibility
49. 49. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. The number 102 is divisible by 3-via the digit sum test. It is divisible by 2 because the last digit is even. Hence 102 is divisible by 2*3 = 6 (102 = 6 *17). Example B. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? How about 2,349,876,543,214 pieces of chocolate with 18 kids? For 384 its digit sum is 15 so it’s divisible by 3 (but not 9). Its last two digits are 84 which is divisible by 4. Hence 384 is divisible by 3 * 4 or 12. Some Facts About Divisibility
50. 50. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. The number 102 is divisible by 3-via the digit sum test. It is divisible by 2 because the last digit is even. Hence 102 is divisible by 2*3 = 6 (102 = 6 *17). Example B. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? How about 2,349,876,543,214 pieces of chocolate with 18 kids? For 384 its digit sum is 15 so it’s divisible by 3 (but not 9). Its last two digits are 84 which is divisible by 4. Hence 384 is divisible by 3 * 4 or 12. For 2,349,876,543,210 for 18, since it's divisible by 2, we only have to test divisibility for 9. Some Facts About Divisibility
51. 51. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. The number 102 is divisible by 3-via the digit sum test. It is divisible by 2 because the last digit is even. Hence 102 is divisible by 2*3 = 6 (102 = 6 *17). Example B. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? How about 2,349,876,543,214 pieces of chocolate with 18 kids? For 384 its digit sum is 15 so it’s divisible by 3 (but not 9). Its last two digits are 84 which is divisible by 4. Hence 384 is divisible by 3 * 4 or 12. Some Facts About Divisibility For 2,349,876,543,210 for 18, since it's divisible by 2, we only have to test divisibility for 9. Instead of actually find the digit sum, let’s cross out the digits sum to multiple of 9.
52. 52. The Multiple Checks Principle If a number passes two different of tests I, II, or III, then it’s divisible by the product of the numbers tested. The number 102 is divisible by 3-via the digit sum test. It is divisible by 2 because the last digit is even. Hence 102 is divisible by 2*3 = 6 (102 = 6 *17). Example B. A bag contains 384 pieces of chocolate, can they be evenly divided by 12 kids? How about 2,349,876,543,214 pieces of chocolate with 18 kids? For 384 its digit sum is 15 so it’s divisible by 3 (but not 9). Its last two digits are 84 which is divisible by 4. Hence 384 is divisible by 3 * 4 or 12. Some Facts About Divisibility For 2,349,876,543,210 for 18, since it's divisible by 2, we only have to test divisibility for 9. Instead of actually find the digit sum, let’s cross out the digits sum to multiple of 9. We see that it’s divisible by 9, hence it’s divisible by 18.
53. 53. Ex. Check each for divisibility by 3 or 6 by inspection. 1. 106 2. 204 3. 402 4. 1134 5. 11340 Check each for divisibility by 4 or 8 by inspection. 6. 116 7. 2040 8. 4020 9. 1096 10. 101340 11. Which numbers in problems 1 – 10 are divisible by 9? Some Facts About Divisibility 12. Which numbers in problems 1 – 10 are divisible by 12? 13. Which numbers in problems 1 – 10 are divisible by 18?