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# 123a-1-f6 some facts about the disvisibility of numbers

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### 123a-1-f6 some facts about the disvisibility of numbers

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