A number “a” is said to be completely divisible by another number “b” if “a” upon divided by the number “b” leaves no remainder. This means that “a” is divisible by “b” if a ÷ b leaves a remainder 0. **Can you determine whether a number is divisible by another number without actually dividing it?**

Here we have some rules, also known as divisibility rules that will help you determine the divisibility of a number without actually going through the entire process of division. These rules can also be referred to as shortcuts to check upon the divisibility of a number.

## Divisibility by 2

**Divisibility Rule – All even numbers are divisible by 2. This means that numbers ending with 0, 2, 4, 6, and 8 are divisible by 2.**

**For example,**

Let us check whether the following numbers are divisible by 2:

202, 305, 638, 7894, 519

Number | Last Digit of the Number | Divisible by 2 |

202 | 2 | Yes |

305 | 5 | No |

638 | 8 | Yes |

7894 | 4 | Yes |

519 | 9 | No |

## Divisibility by 3

**Divisibility Rule – Add the digits of the number whose divisibility by 3 is to be checked. If the sum of these digits is divisible by 3, the number would also be divisible by 3.**

**For example,**

Let us check whether the following numbers are divisible by 3:

345, 796, 6125, 321, 506

Number | Sum of the Digits | Sum Divisible by 3 | Number Divisible by 3 |

345 | 12 | Yes | Yes |

796 | 22 | No | No |

6125 | 14 | No | No |

321 | 6 | Yes | Yes |

506 | 11 | No | No |

# Divisibility by 4

**Divisibility Rule – Check the last two digits of the number. If the number formed by these two digits is divisible by 4, or the last two digits are 00, then the complete number would also be divisible by 4.**

**For example,**

Let us check whether the following numbers are divisible by 4:

488, 564, 3128, 6431, 9147

Number | Last Two Digits | Last Two digits Divisible by 4 | Number Divisible by 4 |

488 | 88 | Yes | Yes |

500 | 00 | Yes | Yes |

3128 | 28 | Yes | Yes |

6431 | 31 | No | No |

9147 | 47 | No | No |

## Divisibility by 5

**Divisibility Rule – Check the units place of the number. A number that ends in 0 or 5 would be divisible by 5.**

**For example,**

Let us check whether the following numbers are divisible by 5:

185, 269, 7846, 490, 355

Number | Number at unit’s place | Number Divisible by 5 |

185 | 5 | Yes |

269 | 9 | No |

7846 | 6 | No |

490 | 0 | Yes |

355 | 5 | Yes |

## Divisibility by 6

**Divisibility Rule – Check whether the given number is divisible by both 2 and 3. If a number is divisible by both 2 and 3, it is also divisible by 6.**

**For example,**

Let us check whether the following numbers are divisible by 6:

618, 948, 4536, 4172, 6541

Number | Number Divisible by 2 | Number divisible by 3 | Number Divisible by 6 |

618 | Yes | Yes | Yes |

948 | Yes | Yes | Yes |

4536 | Yes | Yes | Yes |

4172 | Yes | No | No |

6541 | No | No | No |

## Divisibility by 7

**Divisibility Rule – Check the digit at the unit’s place of the number given. Subtract twice of this digit from the remaining digits. If the number so obtained is divisible by 7, the original number would also be divisible by 7.**

**For example,**

Let us check whether the following numbers are divisible by 7:

749, 357, 5125, 648, 321

Number | Twice of the digit at unit’s place | Number obtained after Subtraction | Resultant Number Divisible by 7 | Original Number Divisible by 7 |

749 | 9 x 2 = 18 | 74 – 18 = 56 | Yes | Yes |

357 | 7 x 2 = 14 | 35 – 14 = 21 | Yes | Yes |

5125 | 5 x 2 = 10 | 512 – 10 = 502 | No | No |

648 | 8 x 2 = 16 | 64 – 16 = 48 | No | No |

321 | 1 x 2 = 2 | 32 – 2 = 30 | Yes | Yes |

## Divisibility by 8

**Divisibility Rule – Check the last three digits of the number. If the number formed by these three digits is divisible by 8, or the last three digits are 000, then the complete number would also be divisible by 8.**

**Also, a number that is divisible by both 2 and 4 will be divisible by 8.**

**For example,**

Let us check whether the following numbers are divisible by 8:

161, 25000, 648, 488, 368

Number | Last Three Digits | Last Three digits Divisible by 8 | Number Divisible by 8 |

1610 | 610 | No | No |

25000 | 000 | Yes | Yes |

20648 | 648 | Yes | Yes |

72488 | 488 | Yes | Yes |

13368 | 368 | Yes | Yes |

# Divisibility by 9

**Divisibility Rule – Add the digits of the number whose divisibility by 9 is to be checked. If the sum of these digits is divisible by 9, the number would also be divisible by 9.**

**For example,**

Let us check whether the following numbers are divisible by 9:

649, 327, 127, 7120, 3025

Number | Sum of the Digits | Sum Divisible by 9 | Number Divisible by 9 |

649 | 19 | No | No |

333 | 18 | Yes | Yes |

129 | 12 | No | No |

7128 | 18 | Yes | Yes |

3025 | 10 | No | No |

## Divisibility by 10

**Divisibility Rule – Check the digit at the unit’s place of the number given. If the number at the unit’s place is 0, the number is divisible by 10.**

Let us check whether the following numbers are divisible by 10:

640, 545, 329, 2148, 9410

Number | Number at unit’s place | Number Divisible by 10 |

640 | 0 | Yes |

545 | 5 | No |

329 | 9 | No |

2148 | 8 | No |

9410 | 0 | Yes |

**It is important to note that apart from the above rules, a number is divisible by a certain number if it is divisible by its factors. For example, a number that is divisible by 2 and 4 would also be divisible by 8. Similarly, a number that is divisible by 2 and 3 will also be divisible by 6.**

**Here are more examples that will help you clearly visualise the above explanations of the divisibility rules and have a better understanding of these short, yet important topics in mathematics.**

Divisibility rules help us work out whether a number is exactly divisible by other numbers (i.e. there is no remainder).

The rules are shortcuts for finding out whether numbers are exactly divisible without doing division calculations. Some of these rules along with examples are illustrated below:

**Divisible by 2?**

Rule: If it ends with a 0, 2, 4, 6, or 8

Number | Divisible? | Why? |

456 | Yes | The last digit is 6 |

68 | Yes | The last digit is 8 |

25 | No | The last digit is 5 (not a 2,4,6,or 8) |

207 | No | The last digit is 7 (not a 2,4,6,or 8) |

**Divisible by 3?**

Rule: If the sum of the digits is a multiple of 3

Number | Divisible? | Why? |

405 | Yes | 4 + 0 + 5 = 9 (9 is a multiple of 3) |

381 | Yes | 3 + 8 + 1 = 12 (12 is a multiple of 3) |

928 | No | 9 + 2 + 8 = 19 (19 is not a multiple of 3) |

4,616 | No | 4 + 6 + 1 + 6 = 17 (17 is not a multiple of 3) |

Helper: The multiples of 3 include… |

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 |

**Divisible by 4?**

Rule: If the last two digits are a multiple of 4

(or if the last two digits are 00)

Number | Divisible? | Why? |

348 | Yes | 48 is a multiple of 4 |

27,616 | Yes | 16 is a multiple of 4 |

8,514 | No | 14 is not a multiple of 4 |

722 | No | 22 is not a multiple of 4 |

1,200 | Yes | The last two digits are 00 200 is a multiple of 4 |

Helper: The multiples of 4 include… |

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 |

**Divisible by 5?**

Rule: If it ends with a 5 or a 0

Number | Divisible? | Why? |

3,425 | Yes | The last digit is 5 |

750 | Yes | The last digit is 0 |

8,551 | No | The last digit is 1 (not a 0 or a 5) |

394 | No | The last digit is 4 (not a 0 or a 5) |

**Divisible by 6?**

Rule: If it is divisible by 2 and by 3

Number | Divisible? | Why? |

5,106 | Yes | The last digit is a 2 (it is a multiple of 2 ) and…5 + 1 + 0 + 6 = 12 (12 is a multiple of 3) |

636 | Yes | The last digit is a 6 (it is a multiple of 2 ) and…6 + 3 + 6 = 15 (15 is a multiple of 3) |

5,912 | No | The last digit is a 2 (it is a multiple of 2 ) but…5 + 9 + 1 + 2 = 17 (17 is not a multiple of 3) |

508 | No | The last digit is a 8 (it is a multiple of 2 ) but…5 + 0 + 8 = 13 (13 is not a multiple of 3) |

**Divisible by 9?**

Rule: If the sum of the digits are a multiple of 9

Number | Divisible? | Why? |

7,686 | Yes | 7 + 6 + 8 + 6 = 27 (27 is a multiple of 9) |

252 | Yes | 2 + 5 + 2 = 9 (9 is a multiple of 9) |

883 | No | 8 + 8 + 3 = 19 (19 is not a multiple of 9) |

5,105 | No | 5 + 1 + 0 + 5 = 11 (11 is not a multiple of 9) |

Helper: The multiples of 9 include… |

9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 |

**Divisible by 10?**

Rule: If the last digit is 0

Number | Divisible? | Why? |

880 | Yes | The last digit is 0 |

9,560 | Yes | The last digit is 0 |

312 | No | The last digit is 2 (not a 0) |

7,897 | No | The last digit is 7 (not a 0) |