# Divisibility Rules

Division, especially of long numbers, can be tedious.  Even when students know all of their times tables and division facts, figuring out if a number is divisible by a certain number can take some time.

Here are some tricks!

0: When working with rational numbers, you cannot divide by 0.  A number divided by 0 is undefined.

1: All numbers are divisible by 1.  Any number divided by 1 is itself.

2: All even numbers can be divided by 2.  Odd numbers cannot be divided evenly by 2.

3: To see if a number is divisible by 3, add the digits.  If the sum of the digits is divisible by 3, the number is divisible by 3.

Is 347 divisible by 3?

3+4+7=14; 14 is not divisible by 3, so 347 is not divisible by 3.

Is 897 divisible by 3?

8+9+7=24; 24 is divisble by 3 so 897 is divisible by 3.

$897 \div 3 = 299$

4: If the number formed by the last 2 digits of any number is divisble by 4, the number is divisble by 4.

Is 342 divisible by 4?

42 is not divisible by 4, so 342 is not divisible by 4.

Is 616 divisible by 4?

16 is divisble by 4 so 616 is divisible by 4.

$616 \div 4 = 154$

5: If a number ends in 0 or 5, it is divisible by 5.

6: If a number is divisible by 2 and 3 (so if it is even and divisible by 3), it is divisible by 6.

7: If you can double the last digit of a number and subtract that from the rest of the number, and the answer is either 0 or divisible by 7, then the number is divisible by 7.

Is 347 divisible by 7?

7+7=14; 34-14= 20; 20 is not divisible by 7, so 347 is not divisible by 7.

Is 182 divisible by 7?

2+2=4; 18-4=14; 14 is divisble by 7 so 182 is divisible by 7.

$182 \div 7 = 26$

Note: with long numbers you can do this several times!

Is 4,795 divisible by 7?

$5\times 2=10$; $479-10=469$;$9\times 2=18$; $46-18=28$ and 28 is divisible by 7, so 4,795 is divisible by 7 too.

$4,795 \div 7=685$

8: If the last three digits of a number form a number that is divisible by 8, then the number is divisible by 8.

Is 12,347 divisible by 8?

347 is not divisible by 8 (8 only goes into even numbers), so 12,347 is not divisible by 8.

Is 876,864 divisible by 8?

864 is divisible by 8 so 876,864 is divisible by 8.

$876,864 \div 8 = 109,608$

9: If the digits of a number add up to a number divisible by 9, the number is divisible by 9.

10: If the last digit of a number is 0, the number is divisible by 10.

11: Alternately subtract then add the digits of a number (start from the left and move right).  If the resulting number is 0 or divisible by 11, the number is divisible by 11.

Is 347 divisible by 11?

3-4+7=6; 6 is not divisible by 11, so 347 is not divisible by 11.

Is 45,892 divisible by 11?

4-5+8-9+2=0; 45,892 is divisible by 11.

$45892 \div 11 = 4172$

12: If a number is divisibe by 3 and 4, then it is divisble by 12.