The ** divisibility criteria ** are those mathematical rules that allow us to discover easily and without the need to solve a division, whether or not a number is divisible by another.

They help us to reduce and simplify the ** fractions**, find the ** Greatest Common Factor** and the ** Least Common Multiple ** of several numbers, decompose any number into ** prime factors**, identify if a number is ** prime ** or ** composite**...

We are going to __ explain __ each of the ** basic rules ** to find the ** multiples and factors** of any natural number using the __ criteria for divisibility from 2 to 15 __

#### Know and apply the divisibility rule of 2

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

__ Example__: 1 3 ** 4 ** ends in ** four ** (even number) and is therefore divisible by 2.

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#### Know and apply the divisibility rule of 3

A ** natural number is divisible by 3 ** if the result of the ** sum of all its digits** is a multiple of 3, that is: 3, 6, 9, 12, 15, 18, 21, 24, 27...

__Example__: ** 3 1 5 ** = 3 + 1 + 5 = ** 9 ** and, therefore, it will be divisible by 3.

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#### Know and apply the divisibility rule of 4

A ** natural number is divisible by 4 ** if it meets either of the two conditions:

__a) Its last two digits are 00__

__Example__: The natural number ** 700 ** is ** divisible ** by ** 4 ** because its last two digits (tens and ones) end in 0.

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__b) Its last two digits are a multiple of 4, for example 04, 08, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72 , 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120...__

__Example__: The number ** 7. 5 3 6 ** is divisible by 4 because its last two digits are ** 36**, which is a multiple of 4 because 36/4 = 9, resulting in an exact division.

#### Know and apply the divisibility rule of 5

A ** natural number is divisible by 5 ** if the value of its last digit (ones place) is ** zero ** or ** five**.

__ Example__: ** 2 2 0 ** ends in ** zero ** and is therefore divisible by 5.

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#### Know and apply the divisibility rule of 6

__ A natural number is divisible by 6 when the divisibility criteria of both 2 and 3 are met simultaneously: __

- A ** natural number is divisible by 2 ** if it ends in ** zero ** or ** even number**, that is, 0, 2, 4, 6 or 8.

__ Example__: 5 8 ** 2 ** ends in ** two ** (even number) and is therefore divisible by 2.

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- A ** natural number is divisible by 3 ** if the result of the ** sum of all its digits** is a multiple of 3, that is: 3, 6, 9, 12, 15, 18, 21, 24, 27...

__ Example__: ** 5 8 2 ** = 5 + 8 + 2 = ** 15 ** and, therefore, it will be divisible by 3.

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Therefore, we can affirm that the number ** 582 is divisible by 6 ** when both the __ divisibility criteria of 2 and 3 are met simultaneously__

#### Know and apply the divisibility rule of 7

A ** natural number is divisible by 7 ** when the difference between the number without the ones digit and twice the ones digit is ** zero ** or a ** multiple of 7**, that is, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147...

__ Example__: ** 5 4 6 ** = 5 4 - ( 2 x 6 ) = 5 4 - 1 2 = 42, which is a multiple of 7 because 42 : 7 = 6 (exact division)

__ Example__: ** 3. 6 8 2 ** = 3 6 8 - ( 2 x 2 ) = 3 6 8 - 4 = 364, which is a multiple of 7 because 364 : 7 = 52 (exact division)

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#### Know and apply the divisibility rule of 8

A ** natural number is divisible by 8 ** if it meets either of the two conditions:

__a) Its last three digits are 000__

__Example__: The natural number ** 14.000 ** is ** divisible ** by ** 8 ** because its last three digits (hundreds, tens, and ones) end in 0.

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__b) Its last three digits are a multiple of 8, for example, 008, 016, 024, 032, 040, 048, 056, 064, 072, 080, 088, 096, 104, 112, 120, 128, 136...__

__Example__: The number ** 6.112 ** is divisible by 8 because its last three digits are ** 112**, which is a multiple of 8 because 112/8 = 14, resulting in an exact division.

#### Know and apply the divisibility rule of 9

A ** natural number is divisible by 9 ** if the result of the __ sum of all its figures is a multiple of 9__, that is: 9, 18, 27, 36, 45, 54, 63, 72, 81...

__ Example__: ** 3. 5 4 6 ** = 3 + 5 + 4 + 6 = ** 18 ** and, therefore, it will be divisible by 9 because 18/9 = 2 (exact division)

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#### Know and apply the divisibility rule of 10

A ** natural number is divisible by 10 ** if its last digit (ones place) is ** zero**.

__ Example__: ** 5 7 0 ** ends in ** zero ** and is therefore divisible by 10.

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#### Know and apply the divisibility rule of 11

A ** natural number is divisible by 11 ** when the sum of the odd position digits, minus the sum of the even position digits, is ** zero ** or a ** multiple of 11**, that is, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187 ...

__Example__: **5 9 . 6 9 7** = ( 5 + 6 + 7 ) - ( 9 + 9 ) = 0

__ Example__: ** 5 6. 7 1 6 ** = (5 + 7 + 6) - (6 + 1) = 18 - 7 = 11, which is a ** multiple of 11 **

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#### Know and apply the divisibility rule of 12

__ A natural number is divisible by 12 when the divisibility criteria of both 3 and 4 are met in parallel: __

- A ** natural number is divisible by 3 ** if the result of the ** sum of all its digits** is a multiple of 3, that is: 3, 6, 9, 12, 15, 18, 21, 24, 27...

__ Example__: ** 4 6 8 ** = 4 + 6 + 8 = ** 18 ** and, therefore, it will be divisible by 3.

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- A ** natural number is divisible by 4 ** if its last two digits are 00 (for example, 1,800) or a two-digit number multiple of 4: 04, 08, 12 , 16, 20, 24, 28, 32, 36, 40, 44, 48, 52...

__ Example__: ** 4 6 8**, its last two digits are ** 68**, which is a multiple of 4 because 68 / 4 = 17 and, therefore, the number ** 468 ** is ** divisible ** by ** 4**.

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Therefore, we can affirm that the number ** 468 is divisible by 12 ** when both the __ divisibility criteria of 3 and 4 are met simultaneously__

#### Know and apply the divisibility rule of 13

A ** natural number is divisible by 13 ** when, when separating the digit from the ones place, multiplying it by 9 and subtracting it from the remaining digits (tens, hundreds, thousands, ten thousands...), the result is the same to 0 or a multiple of 13: 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195...

__Example__: ** 5 4 6 ** = 5 4 - (6 x 9) = 5 4 - 5 4 = ** 0**, therefore __ 546 will be divisible by 13__

__Example__: ** 1. 7 1 6 ** = 1 7 1 - (6 x 9) = 1 7 1 - 5 4 = **117**, which is a multiple of 13 because 117/13 = 9 (exact division) and therefore __ 1,716 will be divisible by 13 __

#### Know and apply the divisibility rule of 15

__ A natural number is divisible by 15 when the divisibility criteria of both 3 and 5 are met synchronously: __

- A ** natural number is divisible by 3 ** if the result of the ** sum of all its digits** is a multiple of 3, that is: 3, 6, 9, 12, 15, 18, 21, 24, 27...

__Example__: ** 3 1 5 ** = 3 + 1 + 5 = ** 9 ** and, therefore, it will be divisible by 3.

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- A ** natural number is divisible by 5 ** if the value of its last digit (ones place) is ** zero ** or ** five**.

__ Example__: ** 3 1 5 ** ends in ** five ** and is therefore divisible by 5.

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Therefore, we can affirm that the number ** 315 is divisible by 15 ** when both the __ divisibility criteria of 3 and 5 are met simultaneously __

#### Find the prime and composite numbers of any natural number

On the other hand, a number is ** composite** when it has __ more than two factors__:

__ Example__: The number ** 10 ** is a ** composite number ** because it can be divided by 1 (the unit), 2, 5 and 10 (by itself)

And remember! The ** number 1 ** is ** neither prime nor compound ** because it only has one divisor, itself.

#### Calculate the multiples of a natural number

We provide you the **lesson** so that you can __ learn what multiples are__ and how to find them from any natural number:

#### Calculate the factors (divisors) of a natural number

We provide you the ** explanation ** so that you can __ learn what factors (divisors) are__ and how to find them from any natural number.

If we want to know what are the ** divisors of any natural number ** __ we would have to divide all the numbers from 1 to the number we want to reach__ (in this exemple, 8) and we would consider ** factors ** those in which the ** division was exact**:

As you may have seen, ** finding the factors of a number can be an expensive task ** if we use larger numbers and, for this reason, we are going to __ teach you the most used method__ to ** calculate the divisors ** of any natural number ** quickly and easily**:

DECOMPOSITION BY FACTORIZATION OF PRIME NUMBERS

#### Find the Least Common Multiple (LCM) of two given natural numbers

Next, we show you the ** two existing procedures ** for ** obtaining the Least Common Multiple ** of two natural numbers: the "__LCM by Listing Method__" and the "__LCM using Prime Factorization__":

LCM OF TWO NUMBERS BY LISTING METHOD

We will find the ** multiples ** of the two natural numbers by following their ** multiplication tables ** in order. The ** Least Common Multiple ** will be the __ smallest of the common multiples__:

LCM OF TWO NUMBERS USING PRIME FACTORIZATION

Starting from two or more natural numbers and through their ** decomposition ** expressed as the ** product of prime factors ** (that is, 2, 3, 5, 7, 11, 13...), the ** Least Common Multiple ** (LCM) will be found by __ multiplying all common and uncommon factors raised to their greatest exponent__:

#### Find the Least Common Multiple (LCM) of three given natural numbers

LCM OF THREE NUMBERS BY LISTING METHOD

We will find the ** multiples ** of the three natural numbers by following their ** multiplication tables ** in order. The ** Least Common Multiple ** will be the __ smallest of the common multiples __:

LCM OF THREE NUMBERS USING PRIME FACTORIZATION

Starting from two or more natural numbers and through their ** decomposition ** expressed as the ** product of prime factors ** (that is, 2, 3, 5, 7, 11, 13...), the ** Least Common Multiple ** (LCM) will be found by __ multiplying all common and uncommon factors raised to their greatest exponent__:

#### Find the Greatest Common Factor (GCF) of two given natural numbers

GCF OF TWO NUMBERS USING PRIME FACTORIZATION

Starting from two or more natural numbers and through their ** decomposition ** expressed as the ** product of prime factors ** (that is, 2, 3, 5, 7, 11, 13...), the ** Greatest Common Factor** (GCF) will be found by __ multiplying all the common factors raised to their lowest exponent__:

The Greek mathematician and geometrist ** Euclid of Alexandria** (325 - 265 BC), also known as the "**father of geometry**", developed an ** algorithm ** to find the ** Greatest Common Factor ** of two natural numbers:

- If when ** dividing the greater number by the smaller number ** the __ division is exact__, the ** GCF ** will be the __ value of the divisor__.

- If when ** dividing the greater number by the smaller number ** the __ division is inexact__, the ** GCF ** will be the __ value of the remainder__.

#### Find the Greatest Common Factor (GCF) of three given natural numbers

GCF OF THREE NUMBERS USING PRIME FACTORIZATION

Starting from three or more natural numbers and through their ** decomposition ** expressed as the ** product of prime factors ** (that is, 2, 3, 5, 7, 11, 13...), the ** Greatest Common Factor** (GCF) will be found by __ multiplying all the common factors raised to their lowest exponent__:

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