For some numbers like two, three, five and others, there are rules that allow you to check divisibility without dividing. These rules are called **divisibility criteria**.

## Divisibility by 2

A natural number is divisible by 2 when it ends in 0, or 2, or 4, or 6, or 8, that is, when it is even.

*Examples:*1) 5040 is divisible by 2 as it ends in 0.

2) 237 is not divisible by 2 because it is not an even number.

## Divisibility by 3

A number is divisible by 3 when the sum of the absolute values of its digits is divisible by 3.

*Example:*234 is divisible by 3, since the sum of its digits is 2 + 3 + 4 = 9, and since 9 is divisible by 3, then 234 is divisible by 3.

## Divisibility by 4

A number is divisible by 4 when it ends in 00 or when the number formed by the last two digits on the right is divisible by 4.

*Example:*1800 is divisible by 4 as it ends in 00.

4116 is divisible by 4, as 16 is divisible by 4.

1324 is divisible by 4, for 24 is divisible by 4.

3850 is not divisible by 4 because it does not end in 00 and 50 is not divisible by 4.

## Divisibility by 5

A natural number is divisible by 5 when it ends in 0 or 5.

*Examples:*1) 55 is divisible by 5, as it ends in 5.

2) 90 is divisible by 5, as it ends in 0.

3) 87 is not divisible by 5, as it does not end in 0 or 5.

## Divisibility by 6

A number is divisible by 6 when it is divisible by 2 and 3.

*Examples:*1) 312 is divisible by 6, because it is divisible by 2 (even) and 3 (sum: 6).

2) 5214 is divisible by 6, because it is divisible by 2 (even) and 3 (sum: 12).

3) 716 is not divisible by 6, (is divisible by 2, but not divisible by 3).

4) 3405 is not divisible by 6 (it is divisible by 3, but not divisible by 2).

## Divisibility by 8

A number is divisible by 8 when it ends in 000, or when the number formed by the last three digits on the right is divisible by 8.

*Examples:*1) 7000 is divisible by 8 as it ends in 000.

2) 56104 is divisible by 8, as 104 is divisible by 8.

3) 61112 is divisible by 8, as 112 is divisible by 8.

4) 78164 is not divisible by 8, as 164 is not divisible by 8.

## Divisibility by 9

A number is divisible by 9 when the sum of the absolute values of its digits is divisible by 9.

*Example:*2871 is divisible by 9, since the sum of its digits is 2 + 8 + 7 + 1 = 18, and since 18 is divisible by 9, then 2871 is divisible by 9.

## Divisibility by 10

A natural number is divisible by 10 when it ends in 0.

*Examples:*

1) 4150 is divisible by 10 as it ends in 0.

2) 2106 is not divisible by 10, as it does not end in 0.

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## Divisibility by 11

A number is divisible by 11 when the difference between the sums of the absolute values of the odd and even order digits is divisible by 11.

The number of the units is 1st order, the second order tens, the third order hundreds, and so on.

*Examples:*

1) 87549

Si (sum of odd orders) = 9 + 5 + 8 = 22

Sp (sum of even orders) = 4 + 7 = 11

Si-Sp = 22-11 = 11

Since 11 is divisible by 11, then the number 87549 is divisible by 11.

2) 439087

Si (sum of odd orders) = 7 + 0 + 3 = 10

Sp (sum of even orders) = 8 + 9 + 4 = 21

Si-Sp = 10-21

Since subtraction cannot be performed, the smallest multiple of 11 (non-zero) is added to the minuendo so that subtraction can be performed: 10 + 11 = 21. Then we have subtraction 21-21 = 0.

Since zero is divisible by 11, the number 439087 is divisible by 11.

## Divisibility by 12

A number is divisible by 12 when it is divisible by 3 and 4.

*Examples:*1) 720 is divisible by 12, because it is divisible by 3 (sum = 9) and 4 (last two digits, 20).

2) 870 is not divisible by 12 (is divisible by 3, but not divisible by 4).

3) 340 is not divisible by 12 (it is divisible by 4, but not divisible by 3).

## Divisibility by 15

A number is divisible by 15 when it is divisible by 3 and 5.

*Examples:*1) 105 is divisible by 15, because it is divisible by 3 (sum = 6) and 5 (ends in 5).

2) 324 is not divisible by 15 (it is divisible by 3, but not divisible by 5).

3) 530 is not divisible by 15 (it is divisible by 5, but not divisible by 3).

## Divisibility by 25

A number is divisible by 25 when the final two digits are 00, 25, 50, or 75.

*Examples:*200, 525, 850 and 975 are divisible by 25.