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Divisibility Rules

Introduction

Numbers that are divisible by one another can be divided to leave a whole number. For example, 33 is divisible by 3, because 333=11 rac{33}{3} = 11.

34 is not divisible by 3 because 343=11.3‾ rac{34}{3} = 11.overline{3}

Divisibility rules are useful shortcuts to work out quickly if one number can be divided by another.

Terms

Divisible by - When one number can be divided by another to make a whole number.
Divisibility rules - Shortcuts that let you know quickly whether a number is divisible by another.

Lesson

Each number has its own test to see if a number is divisible by it. For example, 1 has the simplest divisibility rule of all: is the number an integer. If so, it's divisible by 1.

The divisibility rules for the numbers up to 10 are as follows:

2

If the last digit is even, then the number is divisible by 2.

2398798 is divisible by 2 because 8 is even.

6723643 is NOT divisible by 2, because 3 is odd.

3

If you add up all the digits, and the total is a multiple of three (3,6, or 9), then the number is divisible by 3.

153=1+5+3=9153 = 1+5+3 = 9. Therefore, 153 is divisible by 3.

151=1+5+1=7151 = 1+5+1 = 7. Therefore, 151 is NOT divisible by 3.

If you add up all the digits and get a two digit number, just add those digits together, and repeat until you get a single-digit number.

297=2+9+7=18=1+8=9297 = 2+9+7 = 18 = 1+8 = 9. Therefore 297 is divisible by 3.

295=2+9+5=16=1+6=7295 = 2+9+5 = 16 = 1+6 = 7. Therefore 295 is NOT divisble by 3.

4

If the last two digits are divisible by 4, then the whole number is too.

1716 is divisible by 4, because 16 is a multiple of four.

1718 is NOT divisible by 4, because 18 is not a multiple of 4.

5

If the last digit is 0 or 5 then it is divisible by 5.

8720535 is a multiple of 5.

8720534 is NOT a multiple of 5.

6

To work out if a number is divisible by 6 then check whether the number is even, and then apply the rule for numbers divisible by 3.

150=1+5+0=6150 = 1+5+0 = 6. Therefore, 150 is a multiple of 6, because it is even and a multiple of 3.

153=1+5+3=9153 = 1+5+3 = 9. Therefore, 153 is NOT* a multiple of 6, because even though it is a multiple of 3, it's not an even number.

151=1+5+1=7151 = 1+5+1 = 7. Therefore, 151 is NOT divisible by 6.

7

To work out if a number is divisible by 7, you need to double the last digit and subtract it from the other digits. If this number is a multiple of 7, then the entire number is divisible by 7.

861

Double the last digit: (1 x 2 = 2)

Subtract that from the other digits (86-2 = 84)

84 is a multiple of 7, so 861 is divisible by 7

864

Double the last digit: (4 x 2 = 8)

Subtract that from the other digits (86-8 = 78)

78 is not a multiple of 7, so 864 is NOT divisible by 7

8

If the last three numbers are divisible by 8 then the entire number is divisible by 8. A quick check to see if a number is divisible 8 is to half it, then half it again. If it's still a whole number, then the original number is a multiple of 8.

10416. 416 is a multiple of 8, and therefore 10416 is divisible by 8.

(Half of 416 is 208, and half of that is 104. Since this is a whole number, 416 is a whole number.)

10414. 414 is NOT a multiple of 8, and therefore 10416 is not divisible by 8.

(Half of 414 is 207, and half of that is 103.5, which is not a whole number.)

9

To find if a number is divisible by 9 add up all the digits until you get a single digit number. If the single digit is 9, then the number is divisible by 9. If not, then the number isn't a multiple of 9.

8172=8+1+7+2=18=1+8=98172 = 8+1+7+2 = 18 = 1+8 = 9. Therefore, 8172 is a multiple of 9.

8176=8+1+7+6=22=2+2=48176 = 8+1+7+6 = 22 = 2+2 = 4. Therefore, 8176 is NOT a multiple of 9.

10

If the number ends in 0 it's divisible by 10.

Examples

Which of the following numbers is not divisible by 2?
If a whole number ends in 0, 2, 4, 6, or 8, then the number is called an even number and is divisible by 2.
2,2332,233 (2,2332,233 is not even) →\rightarrow No
6,7726,772 (6,7726,772 is even) →\rightarrow Yes
7,1767,176 (7,1767,176 is even) →\rightarrow Yes
9,0149,014 (9,0149,014 is even) →\rightarrow Yes

Which of the following numbers is not divisible by 3?
If the sum of the digits of a whole number is divisible by 3, then the number itself is divisible by 3.
7,3107,310 (7+3+1+0=11\text{7+3+1+0} = 11, and →\rightarrow/3 = No) No No
1,7221,722 (1+7+2+2=12\text{1+7+2+2} = 12, and →\rightarrow/3 = Yes) Yes Yes
6,4116,411 (6+4+1+1=12\text{6+4+1+1} = 12, and →\rightarrow/3 = Yes) Yes Yes
7,5157,515 (7+5+1+5=18\text{7+5+1+5} = 18, and →\rightarrow/3 = Yes) Yes Yes

Which of the following numbers is not divisible by 4?
If the number represented by the last two digits of a whole number is divisible by 4, then the number itself is divisible by 4.
5,4425,442 (5,442÷4=10.55,442\div4 = 10.5) →\rightarrow No
9,2449,244 (9,244÷4=119,244\div4 = 11) →\rightarrow Yes
5,1725,172 (5,172÷4=185,172\div4 = 18) →\rightarrow Yes
6,0366,036 (6,036÷4=96,036\div4 = 9) →\rightarrow Yes

Which of the following numbers is not divisible by 5?
If a whole number ends in a zero or a 5, then the number is divisible by 5.
2,039→No2,039 \rightarrow \text{No}
5,955→Yes5,955 \rightarrow \text{Yes}
2,635→Yes2,635 \rightarrow \text{Yes}
8,025→Yes8,025 \rightarrow \text{Yes}

Which of the following numbers is not divisible by 6?
If a whole number is divisible by 2 and by 3, then it is divisible by 6.
4,1144,114 (4,1144,114 is even) →\rightarrow Yes
4,1144,114 (4+1+1+4=10\text{4+1+1+4} = 10, and →\rightarrow/3 = No) No No
KaTeX can only parse string typed expression
3,9663,966 (3,9663,966 is even) →\rightarrow Yes
3,9663,966 (3+9+6+6=24\text{3+9+6+6} = 24, and →\rightarrow/3 = Yes) Yes Yes
KaTeX can only parse string typed expression
1,9201,920 (1,9201,920 is even) →\rightarrow Yes
1,9201,920 (1+9+2+0=12\text{1+9+2+0} = 12, and →\rightarrow/3 = Yes) Yes Yes
KaTeX can only parse string typed expression
6,7266,726 (6,7266,726 is even) →\rightarrow Yes
6,7266,726 (6+7+2+6=21\text{6+7+2+6} = 21, and →\rightarrow/3 = Yes) Yes Yes
KaTeX can only parse string typed expression

Which of the following numbers is not divisible by 8?
If the number represented by the last three digits of a whole number is divisible by 8, then the number itself is divisible by 8
2,1812,181 (2,181÷8=22.6252,181\div8 = 22.625) →\rightarrow No
2,2882,288 (2,288÷8=362,288\div8 = 36) →\rightarrow Yes
2,6642,664 (2,664÷8=832,664\div8 = 83) →\rightarrow Yes
5,2005,200 (5,200÷8=255,200\div8 = 25) →\rightarrow Yes

Which of the following numbers is not divisible by 9?
If the sum of the digits of a whole number is divisible by 9, then the number itself is divisible by 9.
5,1495,149 (5+1+4+9=19\text{5+1+4+9} = 19) →\rightarrow No
6,6336,633 (6+6+3+3=18\text{6+6+3+3} = 18) →\rightarrow Yes
2,4932,493 (2+4+9+3=18\text{2+4+9+3} = 18) →\rightarrow Yes
2,2412,241 (2+2+4+1=9\text{2+2+4+1} = 9) →\rightarrow Yes

Divisibility Rules Worksheets (PDF)

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