 # Rules of divisibility

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Any number N which on division by a divisor d gives no remainder (or 0 remainder) is completely divisible by d. In other words, d completely divides N.

## Divisibility test

### Divisibility by 2

Any number which ends with 0, 2, 4, 6 or 8, in other words, is an even number, is divisible by 2.

### Divisibility by 3

A number whose digital root or sum of all digits is divisible by 3 is divisible by 3. For example, 65178 is divisible by 3 and has a digital sum of 6 + 1 + 5 + 7 + 8 = 27 = 2 + 7 = 9, which is divisible by3.

### Divisibility by 4

A number whose last two digits are divisible by 4 is divisible by 4. For example, 83764 is divisible by 4 as 64 is divisible by 4.

### Divisibility by 5

All numbers ending with either 0 or 5 are divisible by 5. For example, 45, 1890, 8475 and 939340 are all divisible by 5.

### Divisibility by 6

As 6 = 3 x 2 Numbers divisible by 3 and 2 are divisible by 6. In other words, all even numbers which are divisible by 3 are divisible by 6.

### Divisibility by 7

Method 1

Progressively subtract two times the last digit from the remaining number. If the result is divisible by 7, then the number is divisible by 7 as well. For example,

3563119 → 356311 - 18 = 35693 → 3569 - 6 = 3563 → 356 - 6 = 35 which is divisible by 7.

Method 2

This method is applicable for very large digits. Assume a number abcdefghij. In this method, the number is divided into blocks of 3 digits each beginning from the right, i.e. a | bcd | efg | hij and alternate blocks are added to give two number N1 = a + efg and N2 = bcd + hij. If the difference of the numbers N1 and N2, i.e. N1 - N2 is divisible by 7, then the number abcdefghij is divisible by 7too.

For example, 6517739025 → 6 | 517 | 739 | 025 → N1 = 745 and N2 = 542 → N1 - N2 = 203 which is divisible by 7.

This property exists as 7 is a factor of 1001. 1001 = 7 x 11 x 13. Therefore, this property is shown by 11 and 13 as well.

### Divisibility by 8

A number whose last three digits are divisible by 8 is divisible by 8. For example, 83864 is divisible by 8 as 864 is divisible by 8.

### Divisibility by 9

A number whose digital root or sum of all digits is divisible by 9 is divisible by 9. For example, 65178 is divisible by 9 and has a digital sum of 6 + 1 + 5 + 7 + 8 = 27 = 2 + 7 = 9,

### Divisibility by 10

Any number which ends in a 0 is divisible by 10.

### Divisibility by 11

To check for divisibility by 11, the difference between the sum of alternate digits of the number must be divisible by 11. Assume a number abcdefghij. Let N1 = a + c + e + g + i and N2 = b + d + f + h + j. If N1 - N2 is divisible by 11, then abcdefghij is divisible by 11. For example, for 1738456038, N1 = 1 + 3 + 4 + 6 + 3 = 17 and N2 = 7 + 8 + 5 + 0 + 8 = 28. As N! - N2 are divisible by 11,1738456038 is divisible by 11.

### Divisibility by 12

All numbers which are divisible by 3 and 4 are divisible by 12.

### Divisibility by 13

Divisibility of 13 is similar to the method 2 mentioned in divisibility of 7. Assume a number abcdefghij. In this method, the number is divided into blocks of 3 digits each beginning from the right, i.e. a | bcd | efg | hij and alternate blocks are added to give two number N1 = a + efg and N2 = bcd + hij. If the difference of the numbers N1 and N2, i.e. N1 - N2 is divisible by 13, then the number abcdefghij is divisible by 13 too.

For example, 6517739020 → 6 | 517 | 739 | 020 → N1 = 745 and N2 = 537 → N1 - N2 = 208 which is divisible by 13.

### Divisibility by 14

All even numbers which are divisible by 7 are divisible by 14.

### Divisibility by 15

Any number ending in 0 or 5 which is also divisible by 3 is divisible by 15.

### Divisibility by 16

All numbers in which the number formed by the last 4 digits is divisible by 16 are divisible by 16.

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