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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
solve it, and find that the product of 6 and 8 is 48.

We say that 8 is a divisor of 48,
because the division 48 ÷ 8
is even (there is no remainder).

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
division or a calculator)
7,854 ÷ 13.

You get 604.153846…  The division
was not even, so 7,854 is NOT
divisible by 13.

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,
so the number is not divisible by 10.

This number IS divisible by 6, since it is 6 times a
number (six times 17).  Remember, if 6 is a factor of a
number, it is also a divisor.

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.

 

Divisible by

2 5 10 100 1000
825          
400          
332          

Divisible by

2 5 10 100 1000
600,200          
56,000          
307,995          

 

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,
add its digits:

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
digits that are divisible by 3 (namely 3, 6, and 9).

For example, to check if 993,768 is divisible by 3,
just add 7 + 8 = 15 and omit 9, 9, 3, and 6. Since
15 is divisible by 3, so is 993,768.

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.
I am divisible by 3 and by 4.
My tens digit is double my ones digit."

Who am I?

“You’ll find me between 110 and 140… 
I don’t end in zero. 
And I am divisible by 12.”



The divisibility rule for 9 is nearly identical to that of the 3:
A number is divisible by 9 if the sum of its digits is divisible by 9.

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.

 

Divisible by

2 3 5 6 9
589          
558          

Divisible by

2 3 5 6 9
495          
3,594          

 

Tip! If you know that a number is divisible by the number n, then you can skip-count by n
 to find more numbers that are also divisible by the number n!

For example, you know that 100 is divisible by 4. Then, 100 − 4 = 96 is also divisible by 4.
Skip-count by fours—up or down—to get a list of numbers that are all 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
divisible by 4, so neither is 5,789.

Why does this work? Because 100 is divisible by 4. So, we already know that 5,700 is
divisible by 4. From then on we just skip-count by 4 to get a list of numbers divisible by 4:
5,704, 5,708, 5,712, 5,716 and so on, which corresponds to 4, 8, 12, 16, and so on.
Therefore, it is enough to look at the last two digits.

10. Mark an "x" if the number is divisible by 2, 3, 4, 5, 6, or 9.

 

Divisible by

2 3 4 5 6 9
1,755            
298            
4,000            
3,270            

Divisible by

2 3 4 5 6 9
3,548            
277            
237            
10,999            

 

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.

 

Divisible by

2 3 4 5 6 8 9
628              
405              

Divisible by

2 3 4 5 6 8 9
938              
224              

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:

18 52 100 502 300 312 348 322
16 44 64 446 292 144 360 422
6 16 72 292 280 266 436 232
86 94 104 144 216 204 568 522
60 54 128 132 244 286 572 588
12 8 12 90 308 312 78 544
15 12 136 98 254 308 348 548
44 48 66 166 256 388 428 444

Divisible by 3:

5 15 23 392 486 500 510 581
3 9 14 298 471 492 501 555
6 21 35 255 444 504 398 577
15 27 39 65 408 354 345 362
17 37 41 99 103 287 285 311
21 33 44 81 88 204 234 254
22 36 51 69 127 171 202 189
9 16 33 72 108 132 156 166

17.

Who am I?

"I am divisible by 8 but not by 5. 
I am greater than 25 but less than 45."

Who am I?

"I am divisible by 6. I am greater than 200
but less than 220. The sum of my digits is 3."




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.

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