Multiple of a number is the result of multiplying that number by any other.

3·1=3 => **3 is multiple of 3**

3·2=6 => **6 is multiple of 3**

3·3=9 => **9 is multiple of 3**

3·4=12 => **12 is multiple of 3**

The products obtained are **multiples of 3**.

Similarly:

Los **multiples of 5** are **5,10,15,20,25,30…**

los **multiples of 7** are **7,14,21,28,35,42…**

and so on…

As we see we can always multiply these numbers by a different number and obtain more multiples, so there are many multiples as numbers!

Any number has an infinite number of multiples.

100 is a multiple of 4 and 25.

All multiples of 100, ie, all numbers ending in two zeros are also multiples of 4 and 25.

To express the multiples of a number we use a dot on the number:

= {5,10,15,20…} it reads like all the multiples of 5.

**HOW TO KNOW IF A NUMBER IS A MULTIPLE OF 2?**

Here is a list of the first 15 **multiples of 2**:

**2, 4, 6, 8,10**

**12,14,16,18,20**

**22,24,26,28,30**

**Just look**: a multiple of 2 always end in even number!(0,2,4,6,8)

If a number ends in 0 or even number, the number is a multiple of 2.

**HOW TO KNOW IF A NUMBER IS A MULTIPLE OF 3?**

Here is a list of the first 10 **multiples of 3**:

** 3, 6, 9,12,15**

**18,21,24,27,30**

**Just look**: If you add up all the digits of a number the result is in the same list!

12 => 1+2 = 3

15 => 1+5 = 6

18 => 1+8 = 9

21 => 2+1 = 3 (start again)

24 => 2+4 = 6

27 => 2+7 = 9

30 => 3+0 = 3 (start again)

If the sum of the digits of a number is 3 or a multiple of 3, the number is a multiple of 3.

The way to explain this is:

Any number formed by the unit followed by zero is equal to a multiple of 3 plus 1

Any number is equal to the sum of a multiple of 3 plus the sum of the numbers.

Then if the sum of its digits is a multiple of 3, the number will also be a multiple of 3.

**HOW TO KNOW IF A NUMBER IS A MULTIPLE OF 4?**

For this example we need a list of larger numbers ** multiples of 4**:

**216, 918, 1032, 40, 512, 100**

**Just look**: Although it is difficult to see, the last two digits are a multiple of 4 or ending in 0!

If the last two digits of a number is a multiple of 4 end in 0 or the number is multiple of 4.

**HOW TO KNOW IF A NUMBER IS A MULTIPLE OF 5?**

Here is a list of the top 10 **multiples of 5**:

**5,10**

**15,20**

**25,30**

**35,40**

**45,50**

**Just look**: A multiple of 5 always end in 0 or 5!

If a number ends in 0 or 5, the number is a multiple of 5.

With this we can extrapolate a rule for multiples of 10, as a multiple of 10 is a multiple of 5 that no ends 5.

So **a 2 digit or more number of 2 that ending in 0 is a multiple of 10. **

**WHAT IS A DIVISOR?**

If one number is a multiple of other number, these number is divisor of the first (and vice versa).

5 | 10 reads like 5 stripes 10

5 is a 10 divisor because 10 is a multiple of 5!

10 is a multiple of 5 because 5 is a 10 divisor

How to know if a number is a divisor of another? (For all we know it’s the opposite to know if is multiple)

**¿Is 5 a 15 divisor?**

We just have to **split**

**15 : 5 = 3**

The division is exact, therefore **yes** **5 is a divisor of 15**, plus 15 is a multiple of 5.

**How do we know the divisors of a number?**

Are those where the number is a multiple of them:

**divisors of 12** = {**1,2,3,4,6,12**}

because 12 is multiple of 1,2,3,4,6,12

**divisors of 13** = {**1,13**} (in this case **13 is a prime number** and just has like divisors the same number and unit)

(To calculate these divisors will do on the issue of **factorization**n).

Como vemos a diferencia de los múltiplos, ¡solo existen un número **limitado de divisores** y esto es porque un número solo puede ser múltiplo de algunos números.

As we see unlike multiples, there are just a **limited number of divisors** of a number and this is because a number can only be a multiple of some numbers.

Any number has a finite number of divisors.

Then:

- When a number is a multiple of another? is the same as asking when the another is divisor of the first

- In addition a number is
**divisible**by other when this other is divisor of this.

Therefore:

**¿Is 10 multiple of 5?** is the same as: ¿is 10 divisible by 5?

10 : 5 = 2 (exact division) then **10 is a multiple of 5**!

**Is 7 multiple of 2?**

7 : 2 = 3,5 (not exact!) then **7 is not multiple of 2**.

**SOME PROPERTIES OF MULTIPLES (consequently divisors)**

Every number is a multiple and divisor himself.

Two numbers can not be multiples of each other at the same time (antisymmetric property)

If a number is a multiple of another, and this in turn is a multiple of another, the first number is a multiple of the third (Transitive property)

The sum of the multiples of a number is also a multiple of that number.

Check this property with multiples of other numbers!

The subtraction of two multiples of a number is also a multiple of that number.

18 – 12 = 6

18 is multiple of 3

12 is multiple of 3

then **6 is multiple of 3**

The product of the multiple of a number by another number is a multiple of the first number.

With all that we have learned, **can you deduce what is a prime number?**