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Divisibility

This is a complete lesson with explanations and exercises about the concept of divisibility, and about factors, divisors, and multiples, meant for fourth grade math. The lesson also reviews the divisibility rules for 2, and 5, and 10.




A number a is divisible by another number b if the division a ÷ b is exact (no remainder).

For example, 18 ÷ 3 = 6. So, 18 is divisible by 3. Also, 18 is divisible by 6, because we can write the other division 18 ÷ 6 = 3. So, 18 is divisible by both 6 and 3.

We say 6 and 3 are divisors or factors of 18.

You can use long division to check if a number is divisible by another.

67 ÷ 4 = 16, R3. There is a remainder, so 67 is not divisible by 4.

Also, from this we learn that neither 4 nor 16 is a factor (divisor) of 67.

1 6
 
4
2 7
4
3

1. Divide and determine if the numbers are divisible by the given number.

a.  21 ÷ 3 = ______

    Is 21 divisible by 3?

 

b. 40 ÷ 6 = _______

Is 40 divisible by 6?

c. 84 ÷ 7 = _______

    Is 7 a factor of 84?

2. Answer the questions. You may need long division.

a.  Is 98 divisible by 4?
 

b.  Is 603 divisible by 7?
 

c.  Is 3 a factor of 1,256?
 



factor factor product
7 × 6 = 42
In any multiplication, the numbers that are multiplied are
called factors and the result is called a  product.

So, since 6 × 7 = 42, 6 and 7 are factors of 42.

From this multiplication fact we can write two divisions:   42 ÷ 6 = 7   and    42 ÷ 7 = 6.
So, this also means that 42 is divisible by both 6 and 7.

Yet one more new word that ties in with all of this: multiple.

We say 42 is a multiple of 6, because 42 is some number times 6, namely 7 × 6.

And of course 42 is also a multiple of 7, because it is some number times 7!

3. Fill in.

    Here's a multiplication fact: 8 × 9 = 72. So, 8 is a ____________________ of 72,
    and so is 9. Also, 72 is a ____________________ of 8, and also 72 is
    a ____________________ of 9. And, 72 is ____________________ by 8
    and also by 9.

4. Fill in.

a.  Is 5 a factor of 55?

Yes, because ____ × ____ = ________.

b.  Is 8 a divisor of 45?

No, because ____ ÷ ____ = _________.

c.  Is 36 a multiple of 6?

____, because ____ × ____ = ______.

d.  Is 34 a multiple of 7?

_____, because ____ ÷ ____ = _______.

e.  Is 7 a factor of 46?

____, because __________________.

f.  Is 63 a multiple of 9?

____, because _________________.

 

Multiples of 6 are all those numbers we get when we multiply 6 by other numbers.
For example, we can multiply 0 × 6, 7 × 6, 11 × 6, 109 × 6, and so on, and the resulting numbers are all multiples of six.

In fact, the skip-counting pattern of 6 gives us a list of multiples of 6:

0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, and so on.

5. a. Make a list of multiples of 11, starting at 0 and at least till 154.

 

    b. Make a list of multiples of 111, starting at 0. Make it as long as you can in this space!



Divisibility by 2

Numbers that are divisible by 2 are called even numbers.
Numbers that are NOT divisible by 2 are called odd numbers.

Even numbers end in 0, 2, 4, 6, or 8. Every second number is even.

Divisibility by 5

Numbers that end in 0 and 5 are divisible by 5.

For example, 10, 35, 720, and 3,675 are such numbers.

6. Mark with “x” if the numbers are divisible by 2 or 5.

number divisible
by 2 by 5
750    
751    
752    
753    
754    
number divisible
by 2 by 5
755    
756    
757    
758    
759    
number divisible
by 2 by 5
760    
761    
762    
763    
764    


Divisibility by 10

Numbers that end in 0 are divisible by 10.

For example, 10, 60, 340, and 2,570 are such numbers.

7. Mark an “x” if the numbers are divisible by 2 or 5 or 10.

number divisible
by 2 by 5 by 10
860      
861      
862      
863      
864      
number divisible
by 2 by 5 by 10

865

     
866      
867      
868      
869      
number divisible
by 2 by 5 by 10
870      
871      
872      
873      
874      

 

If a number is divisible by 10, it ends in zero, so it is ALSO divisible by ____ and ____.


8. a. Write a list of numbers divisible by 2, from 0 to 60.

        _____________________________________________________________

        This is also a list of ______________________________ of 2.

    b. In the list above, underline those numbers that are divisible by 4.
        What do you notice?   

    c. In the list above, color those numbers that are divisible by 6.
        What do you notice?   

    d. Which numbers are divisible by both 4 and 6?
 

9. a. Write a list of numbers divisible by 3, from 0 to 60.

        _____________________________________________________________

        This is also a list of ______________________________ of 3.

    b. In the list above, underline those numbers that are divisible by 6.
        What do you notice?   

    c. In the list above, color those numbers that are divisible by 9.
        What do you notice?   


10. Use the lists you made in (7) and (8). Find numbers that are divisible by both 2 and 9.

 

11. What number is a factor of every number?
 

12. Twenty is a multiple of 4. It is also a multiple of 5. It is also a multiple of four
      other numbers. Which ones?

 

Who am I?
(Hint: I am less than 50.)

Divided by 9, I leave a remainder of 6.
Divided by 4, I leave a remainder of 1.
Divided by 10, I leave a remainder of 3.

Who am I?
(Hint: I am less than 100.)

I am a multiple of 3, 4, 5, and 6.
I am a factor of 120.
Divided by 7, I leave a remainder of 4.




This lesson is taken from Maria Miller's book Math Mammoth Division 2, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.


Math Mammoth Division 2

A self-teaching worktext for 4th grade that covers long division, finding fractional parts with division, word problems, remainder, average, and divisibility.

Download ($5.10). Also available as a printed copy.

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