# Divisibility Rules

Easily test if one number can be exactly divided by another

## Divisible By

"Divisible By" means "when you divide one number by another the result is a whole number"

### Examples:

14 is divisible by 7, because 14 ÷ 7 = 2 exactly

15 is not divisible by 7, because 15 ÷ 7 = 2 17 (the result is not a whole number)

0 is divisible by 7, because 0 ÷ 7 = 0 exactly (0 is a whole number)

"Divisible by" and "can be exactly divided by" mean the same thing

## The Divisibility Rules

These rules let you test if one number is divisible by another, without having to do too much calculation!

### Example: is 723 divisible by 3?

We could try dividing 723 by 3

Or use the "3" rule: 7+2+3=12, and 12 ÷ 3 = 4 exactly  Yes

Note: Zero is divisible by any number (except by itself), so gets a "yes" to all these tests.

1

Any integer (not a fraction) is divisible by 1

2

The last digit is even (0,2,4,6,8)

128  Yes

129  No

3

The sum of the digits is divisible by 3

381 (3+8+1=12, and 12÷3 = 4) Yes

217 (2+1+7=10, and 10÷3 = 3 1/3) No

This rule can be repeated when needed:

99996 (9+9+9+9+6 = 42, then 4+2=6) Yes

4

The last 2 digits are divisible by 4

1312 is (12÷4=3) Yes

7019 is not (19÷4=4 3/4) No

We can also subtract 20 as many times as we want before checking:

68: subtract 3 lots of 20 and we get 8 Yes

102: subtract 5 lots of 20 and we get 2 No

Another method is to halve the number twice and see if the result is still a whole number:

124/2 = 62, 62/2 = 31, and 31 is a whole number. Yes

30/2 = 15, 15/2 = 7.5 which is not a whole number. No

5

The last digit is 0 or 5

175  Yes

809  No

6

Is even and is divisible by 3 (it passes both the 2 rule and 3 rule above)

114 (it is even, and 1+1+4=6 and 6÷3 = 2) Yes

308 (it is even, but 3+0+8=11 and 11÷3 = 3 2/3) No

7

Double the last digit and subtract it from a number made by the other digits. The result must be divisible by 7. (We can apply this rule to that answer again)

672 (Double 2 is 4, 67−4=63, and 63÷7=9) Yes

105 (Double 5 is 10, 10−10=0, and 0 is divisible by 7) Yes

905 (Double 5 is 10, 90−10=80, and 80÷7=11 3/7) No

8

The last three digits are divisible by 8

109816 (816÷8=102) Yes

216302 (302÷8=37 3/4) No

A quick check is to halve three times and the result is still a whole number:

816/2 = 408, 408/2 = 204, 204/2 = 102 Yes

302/2 = 151, 151/2 = 75.5 No

9

The sum of the digits is divisible by 9

(Note: This rule can be repeated when needed)

1629 (1+6+2+9=18, and again, 1+8=9) Yes

2013 (2+0+1+3=6) No

10

The number ends in 0

220  Yes

221  No

11

Add and subtract digits in an alternating pattern (add digit, subtract next digit, add next digit, etc). Then check if that answer is divisible by 11.

1364 (+1−3+6−4 = 0) Yes

913 (+9−1+3 = 11) Yes

3729 (+3−7+2−9 = −11) Yes

987 (+9−8+7 = 8) No

12

The number is divisible by both 3 and 4 (it passes both the 3 rule and 4 rule above)

648
(By 3? 6+4+8=18 and 18÷3=6 Yes)
(By 4? 48÷4=12 Yes)
Both pass, so Yes

524
(By 3? 5+2+4=11, 11÷3= 3 2/3 No)
(Don't need to check by 4) No

There are lots more! Not only are there divisibility tests for larger numbers, but there are more tests for the numbers we have shown.

## Factors Can Be Useful

Factors are the numbers you multiply to get another number: This can be useful, because:

When a number is divisible by another number ...

... then it is also divisible by each of the factors of that number.

Example: If a number is divisible by 6, it is also divisible by 2 and 3

Example: If a number is divisible by 12, it is also divisible by 2, 3, 4 and 6

## Another Rule For 11

• Subtract the last digit from a number made by the other digits.
• If that number is divisible by 11 then the original number is, too.

Can repeat this if needed,

### Example: 286

28 − 6 is 22, which is divisible by 11, so 286 is divisible by 11

### Example: 14641

• 1464 − 1 is 1463
• 146 − 3 is 143
• 14 − 3 is 11, which is divisible by 11, so 14641 is divisible by 11

1625, 1626, 1627, 1628, 2689, 3599, 3600, 3601, 3602, 5007