The document discusses various tests for determining if a number is divisible by certain other numbers based on the digits of the number. Test I examines divisibility by 3 and 9 using the digit sum or digit root of the number. If the digit sum or root is divisible by 3 or 9, then the original number is also divisible. Test II examines divisibility by 2, 4, and 8 based on the last digit(s) of the number - if the last digit is even, it's divisible by 2; if the last 2 digits are divisible by 4, it's divisible by 4; and if the last 3 digits are divisible by 8, it's divisible by 8. Examples are provided to illustrate the application of these tests.

1. We start out with a simple mathematics procedure that is often
used in real live.
Some Facts About Divisibility

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. 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. 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. 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. Some Facts About Divisibility

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. 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. 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. 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. 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. 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. 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. A bag contains 384 pieces of chocolate, can they be evenly
divided by 12 kids?
Some Facts About Divisibility

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. We refer the above digit–sum check for 3 and 9 as test I.
Some Facts About Divisibility

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. The Multiple Checks Principle
Some Facts About Divisibility

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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?