## What is Divisibility?

When we say a number is divisible by another number, we mean that if we divide a whole number by another whole number that the result will be a whole number. For example, $267$ is divisible by $9$, since $267 \div 9 = 29$.

For many numbers it is possible to test for divisibility without actually dividing them. *(video below)*

Some divisibility tests can be done by looking at the ones digit.

A number is:

$\bullet$ divisible by $2$ — if the ones digit is even $(2,4,6,8,0)$

$\bullet$ divisible by $5$ — if the ones digit is $5$ or $0$

$\bullet$ divisible by $10$ — if the ones digit is $0$

Some divisibility tests can be done by adding all the digits together.

A number is:

$\bullet$ divisible by $3$ — if the sum of the digits is divisible by $3$

$\bullet$ divisible by $9$ — if the sum of the digits is divisible by $9$

ex. $87$ is divisible by $3$ because 8+7=15 and 1+5=6 and $6$ is divisible by $3$

ex. $261$ is divisible by $9$ because 2+6+1=9 and $9$ is divisible by $9$

Some less common divisibility tests:

A number is:

$\bullet$ divisible by $4$ — if the last two digits are divisible by $4$

$\bullet$ divisible by $8$ — if the last three digits are divisible by $8$

$\bullet$ divisible by $6$ — if it follows the rules for $2$ and $3$

$\bullet$ divisible by $12$ — if it follows the rules for $3$ and $4$

ex. $432$ is divisible by $3$ because 4+3+2=9 and 9 and $9$ is divisible by $3$

$432$ is divisible by $4$ because $32$ is divisible by $4$

therefore $432$ is divisible by $12$

$\bullet$ divisible by $11$ — if when you add and subtract digits in an alternating pattern, the answer is divisible by $11$.

ex. $1,012$ is divisible by $11$ because +1-0+1-2=0 and $0$ is divisible by $11$

ex. $616$ is divisible by $11$ because +6-1+6=11 and $11$ is divisible by $11$

Check the ** Notes** for these divisible rules and some extras (7, 13 and 17),

and the

**for a connection between divisible rules and Fibonacci numbers.**

*Review*#### Skip to:

- 0:00 Introduction
- 1:00 Divisible by $1$ or $2$
- 1:26 Divisible by $3$
- 1:51 Divisible by $4$
- 2:12 Divisible by $5$
- 2:27 Divisible by $6$
- 2:56 Divisible by $7$
- 3:22 Divisible by $8$
- 3:44 Divisible by $9$
- 4:08 Divisible by $10$
- 4:23 Divisible by $11$
- 5:03 Divisible by $12$
- 5:38 Divisible by $13$
- 7:12 Divisible by $15$
- 7:47 Divisible by $17$