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

2. Some Facts About Divisibility
We start out with a simple mathematics procedure that is often
used in real live. It’s called the digit sum.

3. Some Facts About Divisibility
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.

4. Some Facts About Divisibility
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,

5. Some Facts About Divisibility
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.

6. Some Facts About Divisibility
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
78918273

7. Some Facts About Divisibility
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
add
78918273
15

8. Some Facts About Divisibility
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
add
78918273
15
30

9. Some Facts About Divisibility
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
add
add
78918273
15
30
45

10. Some Facts About Divisibility
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
add
add
78918273
15
30
45
Hence the digit sum of 78999111 is 45.

11. Some Facts About Divisibility
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
add
add
78918273
15
30
45
9
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.

12. Some Facts About Divisibility
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
add
add
78918273
15
30
45
9
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.

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

14. Some Facts About Divisibility
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.

15. Some Facts About Divisibility
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?

16. Some Facts About Divisibility
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?

17. Some Facts About Divisibility
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?
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.

18. Some Facts About Divisibility
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?
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.

19. Some Facts About Divisibility
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?
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.
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).

20. Some Facts About Divisibility
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?
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.
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,

21. Some Facts About Divisibility
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?
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.
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.

22. Some Facts About Divisibility
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?
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.
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,

23. Some Facts About Divisibility
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?
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.
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.

24. Some Facts About Divisibility
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?
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.
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 of those are divisible by 9 by inspection.
a. 2345
b. 356004
c. 6312
d. 870480

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

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

27. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8

28. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8
A number is divisible by 2 if its last digit is even.

29. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8
A number is divisible by 2 if its last digit is even.
A number is divisible by 4 if its last 2 digits are divisible by 4 –
you may ignore all the digits in front of them.

30. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8
A number is divisible by 2 if its last digit is even.
A number is divisible by 4 if its last 2 digits are divisible by 4 –
you may ignore all the digits in front of them.
Hence ****32 is divisible by 4

31. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8
A number is divisible by 2 if its last digit is even.
A number is divisible by 4 if its last 2 digits are divisible by 4 –
you may ignore all the digits in front of them.
Hence ****32 is divisible by 4 but ****42 is not.

32. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8
A number is divisible by 2 if its last digit is even.
A number is divisible by 4 if its last 2 digits are divisible by 4 –
you may ignore all the digits in front of them.
Hence ****32 is divisible by 4 but ****42 is not.
A number is divisible by 8 if its last 3 digits are divisible by 8.

33. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8
A number is divisible by 2 if its last digit is even.
A number is divisible by 4 if its last 2 digits are divisible by 4 –
you may ignore all the digits in front of them.
Hence ****32 is divisible by 4 but ****42 is not.
A number is divisible by 8 if its last 3 digits are divisible by 8.
Hence ****880 is divisible by 8,

34. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8
A number is divisible by 2 if its last digit is even.
A number is divisible by 4 if its last 2 digits are divisible by 4 –
you may ignore all the digits in front of them.
Hence ****32 is divisible by 4 but ****42 is not.
A number is divisible by 8 if its last 3 digits are divisible by 8.
Hence ****880 is divisible by 8, but ****820 is not.

35. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8
A number is divisible by 2 if its last digit is even.
A number is divisible by 4 if its last 2 digits are divisible by 4 –
you may ignore all the digits in front of them.
Hence ****32 is divisible by 4 but ****42 is not.
A number is divisible by 8 if its last 3 digits are divisible by 8.
Hence ****880 is divisible by 8, but ****820 is not.
III. The Test for Divisibility by 5

36. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8
A number is divisible by 2 if its last digit is even.
A number is divisible by 4 if its last 2 digits are divisible by 4 –
you may ignore all the digits in front of them.
Hence ****32 is divisible by 4 but ****42 is not.
A number is divisible by 8 if its last 3 digits are divisible by 8.
Hence ****880 is divisible by 8, but ****820 is not.
III. The Test for Divisibility by 5.
A number is divisible by 5 if its last digit is 5.

37. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8
A number is divisible by 2 if its last digit is even.
A number is divisible by 4 if its last 2 digits are divisible by 4 –
you may ignore all the digits in front of them.
Hence ****32 is divisible by 4 but ****42 is not.
A number is divisible by 8 if its last 3 digits are divisible by 8.
Hence ****880 is divisible by 8, but ****820 is not.
III. The Test for Divisibility by 5.
A number is divisible by 5 if its last digit is 5.
From the above checks, we get the following checks for
important numbers such as 6, 12, 15, 18, 36, etc.

38. Some Facts About Divisibility
We refer the above digit–sum check for 3 and 9 as test I.
We continue with test II and III.
II. The Test for Divisibility by 2, 4, and 8
A number is divisible by 2 if its last digit is even.
A number is divisible by 4 if its last 2 digits are divisible by 4 –
you may ignore all the digits in front of them.
Hence ****32 is divisible by 4 but ****42 is not.
A number is divisible by 8 if its last 3 digits are divisible by 8.
Hence ****880 is divisible by 8, but ****820 is not.
III. The Test for Divisibility by 5.
A number is divisible by 5 if its last digit is 5.
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 number.

39. Some Facts About Divisibility
The Multiple Checks Principle.

40. Some Facts About Divisibility
The Multiple Checks Principle.
If a number passes two different tests I, II, or III, then it’s
divisible by the product of the numbers tested.

41. Some Facts About Divisibility
The Multiple Checks Principle.
If a number passes two different 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.

42. Some Facts About Divisibility
The Multiple Checks Principle.
If a number passes two different 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.

43. Some Facts About Divisibility
The Multiple Checks Principle.
If a number passes two different 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.

44. Some Facts About Divisibility
The Multiple Checks Principle.
If a number passes two different 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).

45. Some Facts About Divisibility
The Multiple Checks Principle.
If a number passes two different 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 2,349,876,543,220
pieces of chocolate with 18 kids?

46. Some Facts About Divisibility
The Multiple Checks Principle.
If a number passes two different 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 2,349,876,543,220
pieces of chocolate with 18 kids?
For 384 its digit sum is 15 so it’s divisible by 3 (but not 9).

47. Some Facts About Divisibility
The Multiple Checks Principle.
If a number passes two different 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 2,349,876,543,220
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.

48. Some Facts About Divisibility
The Multiple Checks Principle.
If a number passes two different 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 2,349,876,543,220
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.

49. Some Facts About Divisibility
The Multiple Checks Principle.
If a number passes two different 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 2,349,876,543,220
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 2,349,876,543,220, since it's divisible by 2,
we only have to test divisibility for 9 to determine if can be
divided by 18 (2 * 9).

50. Some Facts About Divisibility
The Multiple Checks Principle.
If a number passes two different 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 2,349,876,543,220
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 2,349,876,543,220, since it's divisible by 2,
we only have to test divisibility for 9 to determine if can be
divided by 18 (2 * 9). We check its digit sum is 55 which is
not a multiple of 9, so 2,349,876,543,210 is not divisible by 18.

51. Some Facts About Divisibility
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?
12. Which numbers in problems 1 – 10 are divisible by 12?
13. Which numbers in problems 1 – 10 are divisible by 18?