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 *1***7**** (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: 0 (zero) is a "yes" result to any of these tests.*

1

Any integer (not a fraction) is divisible by 1

2

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

12**8**  **Yes**

12**9**  **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

13**12** is (12÷4=3) **Yes**

70**19** is not (19÷4=4 3/4) **No**

A quick check (useful for small numbers) is to halve the number twice and the result is still a whole number.

12/2 = 6, 6/2 = 3, 3 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

17**5**  **Yes**

80**9**  **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

109**816** (816÷8=102) **Yes**

216**302** (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

22**0**  **Yes**

22**1**  **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.

**1**3**6**4 (+1−3+6−4 = **0**) **Yes**

**9**1**3** (+9−1+3 = **11**) **Yes**

**3**7**2**9 (+3−7+2−9 = **−11**) **Yes**

**9**8**7** (+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