Divisibility Tests

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 Review for a connection between divisible rules and Fibonacci numbers.