Divisibility Rules
This is a complete lesson with instruction and exercises about the concept of divisibility and common divisibility rules, meant for 5th or 6th grade. First, it briefly reviews the concepts of factor, divisor, and a number being divisible by another. Then, the "easy" divisibility rules by 2, 5, 10, 100, and 1000 are given. The rest of the lesson concentrates on the divisibility rules by 3, 9, 6, 4, and 8, and has plenty of exercises, including fun labyrinths and mystery number puzzles.
 |
Six is a factor of 48, because 6 × (a number) is 48. The product of 6 and 8 is written 6
× 8. We can also |
We say that
8 is a divisor of 48, |
| The terms factor and divisor mean the same thing. For example, 7 is a divisor of 84 because 84 ÷ 7 is an even division. However, that also means that 7 × (a number) = 84, so 7 is a factor of 84. | ||
1. Answer. In each case, explain why or why not.
a. Is 8 a factor of 100?
b. Is 7 a factor of 3,500?
c. Is 9 a divisor of 50?
|
A number is divisible by another number if the division is even (there is no remainder). |
|
|
Example 1. Is 7,854 divisible by 13? To check, divide
(either by long You get 604.153846… The division |
Example 2. Is 2 × 3 × 17 divisible by 10? By 6? 2 × 3 × 17 is 6 ×
17. The answer to this cannot end in 0, This number IS divisible by 6, since
it is 6 times a |
2. Answer. In each case, explain why or why not (justify your answer).
a. Is 283 divisible by 13?
b. Is 13 × 2,809 divisible by 13?
c. Is 3 × 3 × 3 × 3 × 3 divisible by 2?
d. Is 9,896 divisible by 7?
e. Is 2 × 758 × 5 divisible by 10?
f. Is 2 × 15 × 2 × 7 divisible by 4?
| Easy divisibility rules (You should already know these.) |
|
A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. These are called even numbers. A number is divisible by 5 if it ends in 0 or 5. A number is divisible by 10 if it ends in 0. For example, 56,930 is divisible by 10. A number is divisible by 100 if it ends in “00”. For example, 450,000 is divisible by 100. A number is divisible by 1000 if it ends in "000". For example, 450,000 is divisible by 1000. |
3. Mark with an “x” if the number is divisible by 2, 5, 10, 100, or 1,000.
|
|
|
A number is divisible by 3 if the sum of its digits is divisible by 3. |
|
|
Example. To check if
93,025 is divisible by 3, 9 + 3 + 0 + 2 + 5 = 19 Since 19 is not divisible by 3, neither is 93,025. |
Tip: in adding the
digits, you can totally omit any For example, to check if 993,768 is divisible
by 3, |
4. Are these numbers divisible by 3? If yes, perform the long division and divide the number by 3.
a. 539
b. 43,719
c. 9,032
5. Change one of the digits in the number 238,882
so that it is divisible by 3, but not divisible by 2.
6.
 |
Who am I?
"I am between 50 and 100. |
Who am I? “You’ll find me
between 110 and
140… |
|
The divisibility rule for 9 is nearly identical to that of the 3: |
|
Example. To check if 105,642 is divisible by 9, add its digits: 1 + 0 + 5 + 6 + 4 + 2 = 18 Since 18 is divisible by 9, so is 105,642. |
7. Are these numbers divisible by 9? If yes, perform the long division and divide the number by 9.
a. 888
b. 576
c. 44,082
| A number is divisible by 6 if it is divisible by both 2 and 3. |
8. Mark an "x" if the number is divisible by 2, 3, 5, 6, or 9.
|
|
|
Tip! If you know that a number is divisible by the
number n,
then you can skip-count by n
For example, you know
that 100 is
divisible by 4. Then, 100 − 4
= 96 is also divisible by 4.
100, 96, 92, 88, 84, etc. OR 100, 104, 108, 112, 116, 120, etc. These are consecutive numbers divisible by 4. |
9. a. Make a list of five consecutive numbers
that are divisible by 9, starting
from 99.
b. Make a list of five consecutive numbers that
are divisible by
7, counting down
from 686.
| A number is divisible by 4 if the number formed from its last two digits is divisible by 4. |
|
Example. To check if 5,789 is divisible by 4, just look at the
number 89. It is not |
|
Why does this work? Because 100 is divisible by 4. So, we already
know that 5,700 is |
10. Mark an "x" if the number is divisible by 2, 3, 4, 5, 6, or 9.
|
|
| A number is divisible by 8 if half of it is divisible by 4. |
11. Mark an "x" if the number is divisible by 2, 3, 4, 5, 6, 8, or 9.
|
|
12. Fill in the patterns. Notice the patterns in the remainders!
|
a. 26 ÷ 4 = ______ R ____ 27 ÷ 4 = ______ R ____ 28 ÷ 4 = ______ R ____ 29 ÷ 4 = ______ R ____ 30 ÷ 4 = ______ R ____ 31 ÷ 4 = ______ R ____ 32 ÷ 4 = ______ R ____ |
b.
78 ÷
3 = ______ R ____ 79 ÷ 3 = ______ R ____ 80 ÷ 3 = ______ R ____ 81 ÷ 3 = ______ R ____ 82 ÷ 3 = ______ R ____ 83 ÷ 3 = ______ R ____ 84 ÷ 3 = ______ R ____ |
c. 54 ÷ 7 = ______ R ____ 55 ÷ 7 = ______ R ____ 56 ÷ 7 = ______ R ____ 57 ÷ 7 = ______ R ____ 58 ÷ 7 = ______ R ____ 59 ÷ 7 = ______ R ____ 60 ÷ 7 = ______ R ____ |
13. We know that 686 is evenly divisible by 7.
a. What is the remainder if 687 is divided by 7?
b. What is the remainder if 688 is divided by 7?
c. What is the remainder if 689 is divided by 7?
14. Here is a fact: 1,881 is evenly divisible by 11.
a. What is the remainder if 1,882 is divided by 11?
b. What is the remainder if 1,886 is divided by 11?
c. What is the
remainder if 1,890 is divided by 11?
15. a.
Find a number that is evenly divisible by 6 and is between 90 and 100.
b. Find a number that leaves a remainder of 1 when divided by 6, and is between 90 and 100.
16. Labyrinths!
Find your path from the left to the right side moving right, left,
up, or down (not
diagonally) using numbers that are divisible by the given number. Each
number on your path has
to be greater than the previous number on your path.
|
|
Divisible by 4:
|
Divisible by 3:
|
17.
|
|
Who am I?
"I am divisible
by 8 but not by 5. |
Who am I? "I am divisible by 6.
I am greater
than 200 |
This lesson is taken from Maria Miller's book Math Mammoth Multiplication & Division 3, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.
Math Mammoth Multiplication & Division 3
A self-teaching worktext for 5th grade that covers multi-digit multiplication, long division, problem solving, simple equations, ratios, divisibility, and factoring.
Download ($7.40). Also available as a printed copy.
