Divisibility Tests
The following are some shortcuts for deciding whether a given integer is divisible by or .
Divisibility by
An integer is divisible by if its last digit is or . (In other words, if it's even.)
Example:
is divisible by , since its last digit is .
is not divisible by , since its last digit is not one of or .
Divisibility by
An integer is divisible by if the sum of its digits is divisible by .
Example:
is divisible by , since
,
and is divisible by ( ).
is not divisible by , since
,
and not divisible by .
You can use this rule twice or more times in succession for large numbers. For example, to test for divisibility by , first add the digits:
If you're not sure about , add the digits again.
is definitely divisible by . So, is also.
Divisibility by
An integer is divisible by if the last two digits are divisible by . (This is because goes evenly into .)
Example:
is divisible by , since or is divisible by .
is not divisible by , since is not divisible by .
Divisibility by
An integer is divisible by if the last digit is either or .
Example:
is divisible by , since the last digit is a .
is not divisible by , since the last digit is not a or a .
Divisibility by
An integer is divisible by if it is divisible by and divisible by .
Example:
is divisible by , since:
- It's divisible by (the last digit is )
- It's divisible by ( , which is divisible by ).
is not divisible by , since:
- It's divisible by (the last digit is )
- But it's not divisible by ( , which is not divisible by ).
Divisibility by
Unfortunately, there is no good test for divisibility by .
Divisibility by
An integer is divisible by if the last three digits are divisible by . (This is because goes evenly into .)
Example:
is divisible by , since is divisible by
( ).
is not divisible by , since is not divisible by .
Divisibility by
An integer is divisible by if the sum of the digits is divisible by .
Example:
is divisible by , since:
,
and is divisible by .
is not divisible by , since:
,
and is not divisible by .
Divisibility by
An integer is divisible by if the last digit is
Example:
is divisible by , since the last digit is .
is not divisible by , since the last digit is not .