# 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 $3$ ( $6×3=18$ ).

$2389$ 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 $8$

( $13×8=104$ ).

$29027$ 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 $0$ .