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# Multiples

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## What is a multiple number?

The multiples of a number are the result of the multiplication of the natural number by another natural number, therefore, a number is a multiple of another if it contains an integer number of times.

The multiples of a natural number are infinite, mathematically we can represent as: the multiple of the number is equal to the main number by any natural number.

Multiple of the number = (main number) x (n)

Where n is any natural number.

Mathematically the representation to identify the multiples of a number is by means of the letter “M” in uppercase and then in brackets the number of which we want to know the multiple, we equate to the multiples locked in keys as follows:

M(2) = {0, 2, 4, 6, 8, 10,…, infinite}

The multiples of 2 are obtained by performing the operations 2 x 0 = 0, 2 x 1 = 2, 2 x 2 = 4, 2 x 3 = 6 and so on.

## How do I know if a number is a multiple?

To identify if a number is a multiple of another, the simplest alternative is to make a division between both numbers, verifying that the quotient is an integer (this means that there is no decimal point) and the rest or residue of the division is 0, having the division with the previous characteristics we can conclude that the number is a multiple of the other. In other words, it is only necessary to clear the previous formula and it would remain:

n = Multiple of the number / main number

Where n is any natural number.

## Properties and characteristics of multiples

• Any number is a multiple of itself (checked by multiplying by 1).
• Any number is a multiple of 0 (the result will always be zero).
• Any number is a multiple of 1.
• Any even number is a multiple of 2.
• Any number in which 3, 6 or 9 are added together is a multiple of 3.
• Any number in which the sum of its numbers results in 9 is a multiple of 9.
• Any number whose last digit is 0 or 5 is a multiple of 5.
• Any number whose last digit is 0 is a multiple of 10.

In the "Finding Multiples" section, examples are represented using the above-mentioned properties and characteristics.

## Finding multiples

Using the multiplication table you will find some multiples of some numbers to observe some characteristics.

X 0 1 2 3 4 5 6 7 8 9 10
0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10
2 0 2 4 6 8 10 12 14 16 18 20
3 0 3 6 9 12 15 18 21 24 27 30
4 0 4 8 12 16 20 24 28 32 36 40
5 0 5 10 15 20 25 30 35 40 45 50
6 0 6 12 18 24 30 36 42 48 54 60
7 0 7 14 21 28 35 42 49 56 63 70
8 0 8 16 24 32 40 48 56 64 72 80
9 0 9 18 27 36 45 54 63 72 81 90
10 0 10 20 30 40 50 60 70 80 90 100

In the previous multiplication table the multiplications were made up to the number 10 as a limit, since the multiples are infinite.

### Multiples of 0

M(0) = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…}

Since multiples of 0 always result in 0, it is possible to simplify the representation in the following way:

M(0) = {0}

### Multiples of 1

M(1) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…}

The multiples of 1 always correspond to the multiplied number and according to the properties every number is a multiple of 1.

### Multiples of 2

M(2) = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,…}

The properties of the multiples mentions that every even number is a multiple of 2, that means that every number where the last number is 0, 2, 4, 6 or 8 is a multiple of 2.

### Multiples of 3

M(3) = {0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30,…}

Checking through one of the properties to know the multiples of 3, which indicates that by adding their figures the result can be 3, 6 or 9 is a multiple of 3. For example:

• Number 27: When decomposing the figures 2 + 7 = 9, therefore, 27 is a multiple of 3.
• Number 12: When decomposing the figures 1 + 2 = 3, therefore, 12 is a multiple of 3.
• Number 64: When decomposing the figures: 6 + 4 = 10; Again when decomposing the result: 1 + 0 = 1, therefore, 64 is not a multiple of 3.
• Number 156: When decomposing the figures: 1 + 5 + 6 = 12; Again when decomposing the result: 1 + 2 = 3, therefore, 156 is a multiple of 3.

### Multiples of 4

M(4) = {0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40,…}

### Multiples of 5

M(5) = {0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50,…}

The last number of multiples of 5 ends in 0 or 5

### Multiples of 6

M(6) = {0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60,…}

### Multiples of 7

M(7) = {0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70,…}

### Multiples of 8

M(8) = {0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80,…}

### Multiples of 9

M(9) = {0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90,…}

One of the properties says that adding the figures and the result is 9 would be a multiple of 9.

• Number 63: When decomposing the figures 6 + 3 = 9, therefore, 63 is a multiple of 9.
• Number 765: When decomposing the figures 7 + 6 + 5 = 18; Again we decompose the result: 1 + 8 = 9, therefore, 765 is a multiple of 9.

### Multiples of 10

M(10) = {0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100,…}

The property says that multiples of 10 always end in zero.

## Multiples of two or more numbers

To obtain the common multiples of two or more numbers it is important to know how to obtain the least common multiple (lcm), when obtaining the least common multiple of both numbers, it means that when multiplying by a natural number we are going to obtain the following multiple, for example: lcm x 2, lcm x 3 and so on, the result of the multiplication corresponds to the multiples of the two or more analyzed numbers.

Assuming that we want to find the multiples of 4 and 12, following the procedure to obtain the lcm you get:

4   12  2 2     6  2 1     3  3 1     1

The lcm of 4 and 12 is:

2 x 2 x 3 = 12

This means that the multiples of 4 and 12 are:

M(4, 12) = {0, 12, 24, 48, 60, 72, 84, 96, 108, 120,...}

The multiples of 4 and 12 are obtained by performing the operations 12 x 0, 12 x 1, 12 x 2, 12 x 3 and so on.

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