# Divisibility of natural numbers. Criteria for divisibility

## Criteria for divisibility of natural numbers

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About a natural number *b* we say that it is **divisor ** of a natural number *a* if there is a natural number **c** such that **a** = **b** • **c**.

We can say about *a *that this is a multiple of **b**.

We write **b****|****a** and we read *b divide a* or *b is a divisor of a*.

### Relationship divisibility properties

- For any natural number
*a*, we have *a* | *a*, where* **a* is not null.
- Whatever is the natural number
*a*, then *a* | 0, where *a* not null (not 0) and 1 | *a*.
- Whatever are the natural numbers
*a* and *b*, then *a* | *a•b* and *b | a•b* (the product of a 2 natural numbers is divisibil whith every each factor of the product), where *a* and *b* not null.
- Whatever are the natural numbers
*a, b, c,* if *a | b* and *b | c*, then *a | c*, where *a* and *b* are different from 0.
- Whatever are the natural numbers
*a, b, c*, if *a | b* and *a | c*, then *a | *(*b±c*), where *a* is different from 0.
- Whatever are the natural numbers
*a, b, c*, if *a | b*, then *a|c•b*, where *a* not null.

### Criteria of divisibility

#### The criterion for divisibility by 2

A natural number is divisible by 2 if the last digit is a even figure (0,2,4,6,8).

**E.g.: **16 is divisible by 2 (it has the last digit divisible by 2).

37 is not divisible by 2 (because 7, the last digit, is not divisible by 2).

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#### The criterion for divisibility by 3

A natural number is divisible by 3 if the sum of its digits are divisible by 3.

**E.g.**: 32139 is divisible by 3; 3+2+1+3+9=18

#### The criterion for divisibility by 9

A natural number is divisible by 9 if the sum of its digits are divisible by 9. *This criterion is similar with 3 Criterion*.

**E.g.**. 21543057 is divisible by 9 9; 2+1+5+4+3+0+5+7=27

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#### The criterion for divisibility by 4

A natural number is divisible by 4 if the 2 digits number from the last 2 digits is divisible by 4.

**E.g.**. 4 | 2032 becouse 4 | 32

4 | 128 becouse 4 | 28.

#### The criterion for divisibility by 5

A natural number is divisible by 5 if the last digit is 0 or 5.

**E.g.:** 5 | 35, 5 | 110

#### The criterion for divisibility by 25

A natural number is dicisible by 25 if the number from the last 2 digits is divisible by 25.

**E.g**. 25 | 3850 because 25 | 50

#### The criterion for divisibility by 11

A natural number is divisible by 11 if the difference between the **figures **located **on odd places** and sum of the **digits** located **on even places** is a **multiple of 11**.

**E.g.**: 1925 : 11=175; (9+5)-(1+2)=11

1012 : 11=92; (1+1)-(0+2)=0

#### The criterion of divisibility by 10, 100, 1000, 10.000, 100.000, 1.000.000 etc.

A natural number is divisible by 10 if its last digit is 0,

by 100 if its last two digits are 00,

by 1000 if its last 3 digits are 000,

by 10.000 if its last four digits are 0000,

by 100.000 if its last five digits are 00000,

by 1.000.000 if its last six digits are 000000

and *so on*!

**Keywords: **
divisibility, arithmetic, divisibility criteria, divide, multiply