# Divisibility Tests

The following are some shortcuts for deciding whether a given integer is divisible by $2,3,4,5,6,8,9$ or $10$ .

## Divisibility by $2$

An integer is divisible by $2$ if its last digit is $0,2,4,6$ or $8$ . (In other words, if it's even.)

**
Example:
**

**
$754$
is divisible by
$2$
**
, since its last digit is
$4$
.

**
$8267$
is not divisible by
$2$
**
, since its last digit is not one of
$0,2,4,6$
or
$8$
.

## Divisibility by $3$

An integer is divisible by $3$ if the sum of its digits is divisible by $3$ .

**
Example:
**

**
$747$
is divisible by
$3$
**
, since

$7+4+7=18$ ,

and $18$ is divisible by $3$ ( $6\times 3=18$ ).

**
$2389$
is not divisible by
$3$
**
, since

$2+3+8+9=22$ ,

and $22$ not divisible by $3$ .

You can use this rule twice or more times in succession for large numbers. For example, to test $965787$ for divisibility by $3$ , first add the digits:

$9+6+5+7+8+7=42$

If you're not sure about $42$ , add the digits again.

$4+2=6$

$6$ is definitely divisible by $3$ . So, $965787$ is also.

## Divisibility by $4$

An integer is divisible by $4$ if the last two digits are divisible by $4$ . (This is because $4$ goes evenly into $100$ .)

**
Example:
**

**
$7508$
is divisible by
$4$
**
, since
$08$
or
$8$
is divisible by
$4$
.

**
$8437$
is not divisible by
$4$
**
, since
$37$
is not divisible by
$4$
.

## Divisibility by $5$

An integer is divisible by $5$ if the last digit is either $0$ or $5$ .

**
Example:
**

**
$9375$
is divisible by
$5$
**
, since the last digit is a
$5$
.

**
$8417$
is not divisible by
$5$
**
, since the last digit is not a
$0$
or a
$5$
.

## Divisibility by $6$

An integer is divisible by
$6$
if it is divisible by
$2$
**
and
**
divisible by
$3$
.

**
Example:
**

**
$966$
is divisible by
$6$
**
, since:

- It's divisible by $2$ (the last digit is $6$ )
- It's divisible by $3$ ( $9+6+6=21$ , which is divisible by $3$ ).

**
$268$
is not divisible by
$6$
**
, since:

- It's divisible by $2$ (the last digit is $8$ )
- But it's not divisible by $3$ ( $2+6+8=16$ , which is not divisible by $3$ ).

## Divisibility by $7$

Unfortunately, there is
**
no good test
**
for divisibility by
$7$
.

## Divisibility by $8$

An integer is divisible by $8$ if the last three digits are divisible by $8$ . (This is because $8$ goes evenly into $1000$ .)

**
Example:
**

**
$56104$
is divisible by
$8$
**
, since
$104$
is divisible by
$8$

( $13\times 8=104$ ).

**
$29027$
is not divisible by
$8$
**
, since
$27$
is not divisible by
$8$
.

## Divisibility by $9$

An integer is divisible by $9$ if the sum of the digits is divisible by $9$ .

**
Example:
**

**
$76653$
is divisible by
$9$
**
, since:

$7+6+6+5+3=27$ ,

and $27$ is divisible by $9$ .

**
$29027$
is not divisible by
$9$
**
, since:

$2+9+0+2+7=20$ ,

and $20$ is not divisible by $9$ .

## Divisibility by $10$

An integer is divisible by $10$ if the last digit is $0$

**
Example:
**

**
$46090$
is divisible by
$10$
**
, since the last digit is
$0$
.

**
$29027$
is not divisible by
$10$
**
, since the last digit is not
$0$
.