# Divisibility Rules

Easily test if one number can be evenly 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

But 15 is not divisible by 7, because 15÷7 = 2 1/7 (i.e., the result is not a whole number)

"Divisible by" and "can be evenly 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!

Divisible by: If: Examples:
2 The last digit is even (0,2,4,6,8) 128 is
129 is not
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

4 The last 2 digits are divisible by 4

1312 is (12÷4=3)
7019 is not

5 The last digit is 0 or 5 175 is
809 is not
6 The number is divisible by both 2 and 3 114 (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 If you double the last digit and subtract it from the rest of the number and the answer is:
• 0, or
• divisible by 7
(Note: you can apply this rule to that answer again if you want)

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

9 The sum of the digits is divisible by 9

(Note: you can apply this rule to that answer again if you want)

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 is
221 is not

11

If you sum every second digit and then subtract all other digits and the answer is:

• 0, or
• divisible by 11

1364 ((3+4) - (1+6) = 0) Yes

3729 ((7+9) - (3+2) = 11) Yes

25176 ((5+7) - (2+1+6) = 3) No

12 The number is divisible by both 3 and 4

648
(By 3? 6+4+8=18 and 18÷3=6 Yes.
By 4? 48÷4=12 Yes) 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 first 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