
<Parag>Here you can find arithmetic lessons that will help you understand by explaining in detail and clarifying with some examples.</Parag>


Here you can find arithmetic lessons that will help you understand by explaining in detail and clarifying with some examples.

Arithmetic


# Even And Odd Numbers: Rules And Properties

The set of integers consists of two types of numbers, the even numbers and the odd numbers. In this lesson, we will be able to identify the various properties that distinguish these numbers due to their great importance in the field of mathematics, especially arithmetic and number theory.

Even and odd numbers form the basic building block for understanding how numbers are distributed within the set of integers, as well as a basic starting point for studying prime numbers.

## Even Numbers

An even number is any number that is **divisible** by 2, that is, the **remainder** of dividing this number by 2 is 0.

Here are some examples of even numbers:

<Tex block>\ldots;-6;-4;-2;0;2;4;6;\ldots</Tex>

For the number <Tex>0</Tex>, it is also an even number because <Tex>0\div2=0</Tex>, so the even number rule applies to it.

Now note carefully these even numbers:

<Tex block>
0 \qquad 2 \qquad 4 \qquad 6 \qquad 8 \\\
40 \qquad 42 \qquad 44 \qquad 46 \qquad 48 \\\
90 \qquad 92 \qquad 94 \qquad 96 \qquad 98
</Tex>

You will no doubt have noticed that even numbers always end in one of the digits <Tex>0;2;4;6</Tex> or <Tex>8</Tex> on the far right

### The General Formula For Even Numbers

Let's take the following even numbers:

<Tex block>
\begin{align*}
2 = 2\times 1 \\\
4 = 2\times 2 \\\
6 = 2\times 3 \\\
8 = 2\times 4 \\\
10 = 2\times 5 \\\
\vdots \quad \vdots \quad \vdots \\\
\vdots \quad \vdots \quad \vdots \\\
n = 2\times k
\end{align*}
</Tex>

You can see that an even number can always be decomposed into a product, one of whose factors is 2.

<Note>

We say that <Tex>n</Tex> is an even number if it can be written as:

<Tex block>n=2\cdot k</Tex>

where <Tex>k</Tex> is an integer.

</Note>

## Odd Numbers

An odd Number is any number that is **not divisible** by 2, that is, the **remainder** of dividing this number by 2 is 1.

Here are some examples of odd numbers:

<Tex block>\ldots;-7;-5;-3;-1;1;3;5;7;\ldots</Tex>

Now note carefully these odd numbers:

<Tex block>
1 \qquad 3 \qquad 5 \qquad 7 \qquad 9 \\\
41 \qquad 43 \qquad 45 \qquad 47 \qquad 49 \\\
91 \qquad 93 \qquad 95 \qquad 97 \qquad 99
</Tex>

You will no doubt have noticed that odd numbers always end in one of the digits <Tex>1;3;5;7</Tex> or <Tex>9</Tex> on the far right

## The General Formula For Odd Numbers

Let's take the following odd numbers:

<Tex block>
\begin{align*}
1 = 2\times 0+1 \\\
3 = 2\times 1+1 \\\
5 = 2\times 2+1 \\\
7 = 2\times 3+1 \\\
9 = 2\times 4+1 \\\
\vdots \quad \vdots \quad \vdots \quad \vdots \\\
\vdots \quad \vdots \quad \vdots \quad \vdots \\\
n = 2\times k+1
\end{align*}
</Tex>

You can see that an odd number can always be decomposed into a sum of 1 and an even number.

<Note>
We say that <Tex>n</Tex> is an odd number if it can be written as:

<Tex block>n=2\cdot k+1</Tex>

where <Tex>k</Tex> is an integer.

</Note>

<Example type="example" num="1"/>

Write the following odd numbers as odd number form: <Tex>103; -65; 1001</Tex>

**Solution**

The form of odd number is <Tex>n=2k+1</Tex>, so we must fin the integer <Tex>k</Tex>.

<Tex block>
103=2\times51+1 \\\
-65=2\times(-33)+1 \\\
1001=2\times500+1
</Tex>

## Properties of Even and Odd Numbers

In this section, we will try to identify some important properties of even and odd numbers through the use of some algebraic properties. Each property we will demonstrate with a simple proof.

### Property 1:

<Note>
The sum of two even numbers is an even number.

</Note>

**Proof**

Let <Tex>n</Tex> and <Tex>m</Tex> are two even numbers. So:

<Tex blok>
\qquad n=2k \\\
\qquad m=2k'
</Tex>

Put the two equations together side by side, and use the distributive property:

<Tex block>
n+m=2k+2k' \\\
n+m=2(k+k') \\\
n+m=2k"
</Tex>

So we get the sum <Tex>2k"</Tex>  written in even number form.

<Example type="example" num="2"/>

The sum of the two even numbers 12 and 16 is the even number 28:

<Tex block>12+16=28=2\cdot14</Tex>

### Property 2:

<Note>
The sum of two odd numbers is an even number.

</Note>

**Proof**

Let <Tex>n</Tex> and <Tex>m</Tex> are two odd numbers. So:

<Tex blok>
\qquad n=2k+1 \\\
\qquad m=2k'+1
</Tex>

Put the two equations together side by side, and use the distributive property:

<Tex block>
n+m=2k+1+2k'+1 \\\
n+m=2k+2k'+2 \\\
n+m=2(k+k'+1) \\\
n+m=2k"
</Tex>

So we get the sum <Tex>2k"</Tex>  written in even number form.

<Example type="example" num="3"/>

The sum of the two even numbers 11 and 19 is the even number 20:

<Tex block>11+9=20=2\cdot10</Tex>

### Property 3:

<Note>
The sum of an odd number and an even number is an odd number.

</Note>

**Proof**

Let <Tex>n</Tex> an even number and <Tex>m</Tex> an odd number. So:

<Tex blok>
\qquad n=2k \\\
\qquad m=2k'+1
</Tex>

Put the two equations together side by side, and use the distributive property:

<Tex block>
n+m=2k+2k'+1 \\\
n+m=2k+2k'+1 \\\
n+m=2(k+k')+1 \\\
n+m=2k"+1
</Tex>

So we get the sum <Tex>2k"+1</Tex>  written in odd number form.

<Example type="example" num="4"/>

The sum of the even number 20 and the odd number 17 is the odd number 37:

<Tex block>20+17=37=2\cdot18+1</Tex>

### Property 4:

<Note>
The product of two even numbers is an even number.

</Note>

**Proof**

Let <Tex>n</Tex> and <Tex>m</Tex> are two even numbers. So:

<Tex blok>
\qquad n=2k \\\
\qquad m=2k'
</Tex>

Multiply the two equations side by side, and use the distributive property:

<Tex block>
n\times m=2k\times 2k' \\\
n\times m=4kk' \\\
n\times m=2(2kk') \\\
n\times m=2k"
</Tex>

So we get the product <Tex>2k"</Tex> written in even number form.

<Example type="example" num="5"/>

The product of the two even numbers 12 and 4 is the even number 48:

<Tex block>12\times4=48=2\cdot24</Tex>

### Property 5:

<Note>
The product of two odd numbers is an odd number.

</Note>

**Proof**

Let <Tex>n</Tex> and <Tex>m</Tex> are two odd numbers. So:

<Tex blok>
\qquad n=2k+1 \\\
\qquad m=2k'+1
</Tex>

Multiply the two equations together side by side, and use the distributive property:

<Tex block>
n\times m=(2k+1)\times (2k'+1) \\\
n\times m=4kk'+2k+2k'+1 \\\
n\times m=2(2kk'k+k')+1 \\\
n\times m=2k"+1
</Tex>

So we get the sum <Tex>2k"+1</Tex> written in odd number form.

<Example type="example" num="6"/>

The product of the two odd numbers 11 and 7 is the odd number 77:

<Tex block>11\times7=77=2\cdot38+1</Tex>

### Property 6:

<Note>
The product of an odd number and an even number is an even number.

</Note>

**Proof**

Let <Tex>n</Tex> an even number and <Tex>m</Tex> an odd number. So:

<Tex blok>
\qquad n=2k \\\
\qquad m=2k'+1
</Tex>

Multiply the two equations together side by side, and use the distributive property:

<Tex block>
n\times m=2k\times (2k'+1) \\\
n\times m=4kk'+2k \\\
n\times m=2(2kk'+k) \\\
n\times m=2k"
</Tex>

So we get the sum <Tex>2k"</Tex>  written in even number form.

<Example type="example" num="4"/>

The sum of the odd number 5 and the even number 10 is the even number 50:

<Tex block>5\times10=20=2\cdot10</Tex>

<Related>

<Link to="../arithmetic/multiples-divisors-whole-numbers" as="/lessons/arithmetic/multiples-divisors-whole-numbers">Multiples and divisors of Whole Numbers</Link>

</Related>

Even and Odd Numbers and important rules and properties, general formula, the sum and the product of even and odd numbers

Even And Odd Numbers Rules


# How To Find Tne Multiples (Factors) And The Divisors Of A Whole Number

In this lesson, we will learn how to determine the multiples and divisors of a natural integer, and the relation between them. An easy and important lesson for studying arithmetic lessons. We hope you benefit from our lesson.

## Divisors Of A Natural Integer

Let's take the number 24, for example. If we divide the number 24 by the numbers from 1 to 24, we will notice that the result is always either a natural integer (the remainder is equal to 0) or a non natural number (meaning the remainder is different from 0).

Let's do this division:

<Tex block>
\begin{align*}
24 \div 1 &= 24 \longrightarrow \\;1\\; is\\; divisor\\; of\\; 24\\\
24 \div 2 &= 12 \longrightarrow \\;2\\; is\\; divisor\\; of\\; 24\\\
24 \div 3 &= 8 \longrightarrow \\;3\\; is\\; divisor\\; of\\; 24\\\
24 \div 4 &= 6 \longrightarrow \\;4\\; is\\; divisor\\; of\\; 24\\\
24 \div 5 &= 4,8 \\\
24 \div 6 &= 4 \longrightarrow \\;6\\; is\\; divisor\\; of\\; 24\\\
24 \div 7 &= 3,428571429 \\\
24 \div 8 &= 3 \longrightarrow \\;8\\; is\\; divisor\\; of\\; 24\\\
24 \div 9 &= 2,66666666 \\\
24 \div 10 &= 2,4 \\\
24 \div 11 &= 2,18181818 \\\
24 \div 12 &= 2 \longrightarrow \\;12\\; is\\; divisor\\; of\\; 24\\\
24 \div 13 &= 1,84615384 \\\
24 \div 14 &= 1,71428571 \\\
24 \div 15 &= 1,6 \\\
24 \div 16 &= 1,5 \\\
24 \div 17 &= 1,411764706 \\\
24 \div 18 &= 1,333333333 \\\
24 \div 19 &= 1,263157895 \\\
24 \div 20 &= 1,2 \\\
24 \div 21 &= 1,142857143 \\\
24 \div 22 &= 1,09090909 \\\
24 \div 23 &= 1,04347826 \\\
24 \div 24 &= 1 \longrightarrow \\;24\\; is\\; divisor\\; of\\; 24\\\
\end{align*}
</Tex>

So the divisors of 24 are the following numbers: <Tex>\\{1;\\;2;\\;3;\\;4;\\;6;\\;8;\\;12;\\;24 \\}</Tex>, and is not divisible by the others.

<Image src="/imgs/lessons/arithmetic/divisor.png" alt="Dividend, Divisor and Quotien" width='500px' height='205px' />

- The number 24 is called: **Dividend**
- The number 8 is called: **Divisor**
- The number 3 is called: **Quotient**

So this property can be summarized as follows:

<Note>
Let <Tex>a</Tex>, <Tex>d</Tex> and <Tex>q</Tex> are three whole numbers with <Tex>d</Tex> is non null.
The number <Tex>d</Tex> is a divisor of the number <Tex>a</Tex> if the remainder is equal to 0.
<Tex block>a \div d = q</Tex>
</Note>

If the number <Tex>a</Tex> accepts only two divisors, 1 and itself, we say that the number <Tex>a</Tex> is a  [prime number.](https://en.wikipedia.org/wiki/Prime_number)

<Example type="example" num="1"/>
Find the divisors of each of the following numbers: 27; 29; 48; 72 and 100.

<Example type="solution"/>

 The divisors of 27 are: <Tex>\\{1;\\; 3;\\; 9;\\; 27 \\}.</Tex>

 The divisors of 29 are: <Tex>\\{1;\\; 29 \\}</Tex>, so, 29 is a prime number.

 The divisors of 48 are: <Tex>\\{1;\\; 2;\\; 3;\\; 4;\\; 6;\\; 8;\\; 12;\\; 16;\\; 24;\\; 48 \\}.</Tex>

 The divisors of 72 are: <Tex>\\{1;\\; 2;\\; 3;\\; 4;\\; 6;\\; 8;\\; 9;\\; 12;\\; 18;\\; 24;\\; 36;\\; 72 \\}.</Tex>

 The divisors of 100 are: <Tex>\\{1;\\; 2;\\; 4;\\; 5;\\; 10;\\; 20;\\; 25;\\ 50;\\; 100 \\}.</Tex>

 ## Multiples Of A Natural Integer

 For example, let's take the number 8 and determine its multiples. To find multiples of 8, we multiply 8 by 1, then by 2, then by 3, and so on. We get an infinite set of multiples.

To be more clear, we have:

<Tex block>
\begin{align*}
8 \times 1 &=8 \\\
8 \times 2 &=16 \\\
8 \times 3 &=24 \\\
8 \times 4 &=32 \\\
\vdots \quad &\vdots \quad \vdots
\end{align*}
</Tex>

So as you can see, the multiples of 8 are the set: <Tex>\\{1;\\; 8;\\; 16;\\; 24;\\; 32; \ldots \\}</Tex>, where we get the current multiplier by adding 8 to the multiplier before it.

<Image src="/imgs/lessons/arithmetic/multiple.png" alt="factor1, factor2 and multiple" width='500px' height='205px' />

- The number 8 is called: **Factor 1**
- The number 3 is called: **Factor 2**
- The number 24 is the **multiple** of the two factors 3 and 8.

So this property can be summarized as follows:

<Note>

 Let <Tex>a</Tex>, <Tex>b</Tex> and <Tex>c</Tex> three natural integers.

 <Tex>c</Tex> is the multiple of <Tex>b</Tex> if :

 <Tex block>a \times b = c</Tex>

</Note>

## Special Cases Of Divisors And Multiples

- 1 has only one divisor that is itself
- The divisors of 0 are all whole numbers except 0.
- The multiples of 1 are all whole numbers.
- 0 has only one multiple that is itself.

<Example type="example" num="2"/>
Find the multiples less than 60 of each of the following numbers: 6; 7; 9; 11 and 20.

<Example type="solution"/>

The multiples less than 60 of 6 are: <Tex>\\{6;\\; 12;\\; 18;\\; 24;\\; 30;\\; 36;\\; 42;\\; 48;\\; 54 \\}.</Tex>

The multiples less than 60 of 7 are: <Tex>\\{7;\\; 14;\\; 21;\\; 28;\\; 35;\\; 42;\\; 49;\\; 56 \\}.</Tex>

The multiples less than 60 of 9 are: <Tex>\\{9;\\; 18;\\; 27;\\; 36;\\; 45;\\; 54 \\}.</Tex>

The multiples less than 60 of 11 are: <Tex>\\{11;\\; 22;\\; 33;\\; 44;\\; 55 \\}.</Tex>

The multiples less than 60 of 20 are: <Tex>\\{20;\\; 40 \\}.</Tex>

## The Relation between Divisors And Multiples

Let's go back to the example we took in the divisors section. The divisors of 24 are: <Tex>\\{1;\\;2;\\;3;\\;4;\\;6;\\;8;\\;12;\\;24 \\}</Tex>.

And the example we took in the multiples section. The multiples of 8 are: <Tex>\\{1;\\; 8;\\; 16;\\; 24;\\; 32; \ldots \\}</Tex>.

We notice that the number 8 is a divisor of 24, and the number 24 is a multiple of 8.

<Note>

If <Tex>a</Tex> is a divisor of <Tex>b</Tex>, then, <Tex>b</Tex> is a multiple of <Tex>a</Tex>.

</Note>

<Related>

<Link to="../arithmetic/even-odd-numbers-rules" as="/lessons/arithmetic/even-odd-numbers-rules">Even And Odd Numbers Rules</Link>

</Related>


Learn how to find the divisors and the multiples (factors) of whole numbers and the relation between multiples and divisors.

Multiples and divisors of Whole Numbers


# Finding Greatest Common Factor (GCF) And Least Common Multiple (LCM)

Among the important concepts in arithmetic that a student of mathematics will address, we find the concepts of the greatest common divisor and the least common multiple of two natural integers.
To understand how to calculate them, you must first understand how to determine <Link to="../arithmetic/multiples-divisors-whole-numbers" as="/lessons/arithmetic/multiples-divisors-whole-numbers">the factors and multiples of a natural integer</Link>.

## Greatest Common Factor (Divisor)

Let's take, for example, the numbers 18 and 24. To determine the greatest common divisor, we first determine the denominators of 18 and the divisors of 24 as follows:

- Divisors of 18 are: <Tex>\\{ 1;\\; 2;\\;3;\\; 6;\\; 9;\\; 18 \\}</Tex>
- Divisors of 24 are: <Tex>\\{ 1;\\; 2;\\; 3;\\; 4;\\; 6;\\; 8;\\; 12;\\; 24 \\}</Tex>

We note that there are a set of common divisors: <Tex>\\{ 1;\\; 2;\\; 3\\; 6 \\}</Tex>

The greatest of this divisors is the greatest common divisor, and we write <Tex>gcd(18;24)=6</Tex>.

<Note>

The greatest common divisor of two natural integers is the largest number that divides these two numbers.

in other word, let <Tex>a</Tex>, <Tex>b</Tex> and <Tex>d</Tex> three non-zero natural integers.

If <Tex>gcd(a;b)=d</Tex> therefore, there exist two natural integers k and k' such that <Tex>a=d \times k</Tex> and <Tex>b=d \times k'</Tex>.

</Note>

<Alert>

No doubt you have noticed that once we say "Greatest Common Factor" and once we say "Greatest Common Divisor", they are in fact the same thing.

- Greatest Common Factor (GCF)
- Greatest Common Divisor (GCD)
- Highest Common factor (HCF)

</Alert>

### Greatest common factor for more than two numbers

In the previous paragraph, we determined the gcd of two numbers only because it is the most used in arithmetic lessons, but in some cases we have to calculate the greatest common divisor of more than two numbers. In the same way, we display the list of divisors of each number and then take the greatest common divisor.

For example, let's define gcd of the numbers 18, 20 and 28.

- Divisors of 18 are: <Tex>\\{ 1;\\; 2;\\;3;\\; 6;\\; 9;\\; 18 \\}</Tex>
- Divisors of 20 are: <Tex>\\{ 1;\\; 2;\\; 4;\\; 5;\\; 10;\\; 20 \\}</Tex>
- Divisors of 28 are: <Tex>\\{ 1;\\; 2;\\; 4;\\; 7;\\; 14;\\; 28 \\}</Tex>

We note that there are a set of common divisors: <Tex>\\{ 1;\\; 2 \\}</Tex>, therefore, <Tex>gcd(18;20;28)=2</Tex>

<Example type="example" num="1"/>

Find: <Tex>gcd(36;16)</Tex>; <Tex>gcd(96;64;48)</Tex> and <Tex>gcd(9;8)</Tex>.

<Example type="solution"/>

After we've done some examples, now you can easily find gcd without going back every time to all the steps in the previous examples. From now on, you will be able to find gcd more quickly just by looking at the two numbers and working out their factors mentally.

- Finding <Tex>gcd(36;16)</Tex>

    Divisors of 36: <Tex>\\{ 1;\\; 2;\\; 3;\\; 4;\\; 6;\\; 9;\\; 12;\\ 18;\\; 36 \\}</Tex>

    Divisors of 16 are: <Tex>\\{ 1;\\; 2;\\;4;\\; 8;\\; 16 \\}</Tex>

    Thus, <Tex>gcd(36;16)=4</Tex>

- Finding <Tex>gcd(96;64;48)</Tex>

    Divisors of 96: <Tex>\\{ 1;\\; 2;\\; 3;\\; 4;\\; 6;\\; 8;\\; 12;\\ 16;\\; 24;\\; 32;\\; 48;\\; 96 \\}</Tex>

    Divisors of 64 are: <Tex>\\{ 1;\\; 2;\\;4;\\; 8;\\; 16;\\; 32;\\; 64 \\}</Tex>

    Divisors of 48: <Tex>\\{ 1;\\; 2;\\; 3;\\; 4;\\; 6;\\; 8;\\; 12;\\ 16;\\; 24;\\; 48 \\}</Tex>

    Thus, <Tex>gcd(96;64;48)=16</Tex>

- Finding <Tex>gcd(9;8)</Tex>

    Divisors of 9: <Tex>\\{ 1;\\; 3;\\; 9 \\}</Tex>

    Divisors of 8 are: <Tex>\\{ 1;\\; 2;\\;4;\\; 8 \\}</Tex>

    Thus, <Tex>gcd(9;8)=1</Tex>

<Alert>

if the greatest common divisor of <Tex>a</Tex> and <Tex>b</Tex> is 1, we say that <Tex>a</Tex> and <Tex>b</Tex> are [coprime](https://en.wikipedia.org/wiki/Coprime_integers).

</Alert>

## Least Common Multiple (LCM)

As with the greatest common divisor, we determine the multiples of the two numbers and then compare the common multiples and take the smallest of them, which will be the least common multiple (lcm).

Let's take, for example, the numbers 4 and 6. 

- Multiples of 4 are: <Tex>\\{ 4;\\; 8;\\; 12;\\; 16;\\; 20;\\; 24;\\; \cdots \\}</Tex>
- Multiples of 6 are: <Tex>\\{ 6;\\; 12;\\; 18;\\; 24;\\; 30;\\; 36;\\; \cdots \\}</Tex>

So the common multiples are all numbers that are divisible by 4 and 6 at the same time and are an infinite set. In this example, 12 is the smallest common multiple of 4 and 6, and no other number smaller than 12 is divisible by 4 and 6 at the same time.

Thus we write: <Tex>lcm(4;6)=12</Tex>

<Note>

The least common multiple of two numbers is the smallest number that is divisible by those two numbers.

Let <Tex>a</Tex>, <Tex>b</Tex> and <Tex>c</Tex> three non-zero natural integers.

if <Tex>lcm(a;b)=c</Tex> then, there exist two natural integers k and k' such that <Tex>k</Tex> and <Tex>k'</Tex>  such that: <Tex>c \div a = k</Tex> and <Tex>c \div b = k'</Tex>.

</Note>

<Example type="example" num="2"/>

Find: <Tex>lcm(7;4)</Tex> and <Tex>lcm(2;3;5)</Tex>.

<Example type="solution"/>

- Finding <Tex>lcm(7;4)</Tex>

    Multiple of 7 are: <Tex>\\{ 7;\\; 14;\\; 21;\\; 28;\\; 35;\\; 42;\\; \cdots \\}</Tex>

    Multiples of 4 are: <Tex>\\{ 4;\\; 8;\\; 12;\\; 16;\\; 20;\\; 24;\\; 28;\\; 32;\\; \cdots \\}</Tex>

    Thus, <Tex>lcm(7;4)=28</Tex>

- Finding <Tex>lcm(2;3;5)</Tex>

    Multiple of 2 are: <Tex>\\{ 2;\\; 4;\\; 6;\\; 8;\\; 10;\\; 12;\\; 14;\\; 16;\\; 18;\\; 20;\\; 22;\\; 24;\\; 26;\\; 28;\\; 30;\\; \cdots \\}</Tex>

    Multiples of 3 are: <Tex>\\{ 3;\\; 6;\\; 9;\\; 12;\\; 15;\\; 18;\\; 21;\\; 24;\\; 27;\\; 30;\\; 33;\\; \cdots \\}</Tex>

    Multiples of 5 are: <Tex>\\{ 5;\\; 10;\\; 15;\\; 20;\\; 25;\\; 30;\\; 35;\\; 40;\\; \cdots \\}</Tex>

    Thus, <Tex>lcm(2;3;5)=30</Tex>

## Relationship between LCM and GCD

Let's find the <Tex>gcd(6;15)</Tex> and <Tex>lcm(6;15)</Tex>.

- Finding <Tex>gcd(6;15)</Tex>

    Divisors of 6 are: <Tex>\\{ 1;\\; 2;\\; 3;\\; 6 \\}</Tex>

    Divisor of 15 are: <Tex>\\{ 1;\\;3; \\; 5;\\; 15 \\}</Tex>

    Thus, <Tex>gcd(6;15)=3</Tex>

- Finding <Tex>lcm(6;15)</Tex>

    Multiples of 6 are: <Tex>\\{ 6; \\;12; 18;\\; 24;\\; 30;\\; 36;\\; \cdots \\}</Tex>

    Multiples of 15 are: <Tex>\\{ 15; \\;30;\\; 45;\\; \cdots \\}</Tex>

    Thus, <Tex>lcm(6;15)=30</Tex>

Can you infer the relationship between the gcd and the lcm in this example?

If you observe carefully, you will find that the product of the gcd and the lcm is equal to the product of 6 and 15.

<Tex>
\begin{cases}
gcd(6;15) \times lcm(6;15) = 3 \times 30 =90 \\\
6 \times 15 =90
\end{cases}
</Tex>

Thus, <Tex>gcd(6;15) \times lcm(6;15) = 6\times15</Tex>

<Note>

Let <Tex>a</Tex> and <Tex>b</Tex> two non-zero natural integers.

<Tex block>gcd(a;b) \times lcm(a;b) = a \times b</Tex>

</Note>

<Example type="example" num="3"/>

Calculate <Tex>lcm(48;84)</Tex> knowing that <Tex>gcd(48;84)=12</Tex>.

<Example type="solution"/>

We know that: <Tex>gcd(48;84) \times lcm(48;84) = 48 \times 84</Tex>

Thus,

<Tex block>
\begin{align*}
lcm(48;84)&=\frac{48 \times 84}{gcd(48;84)} \\\
&= \frac{4032}{12} \\\
&= 336
\end{align*}
</Tex>

Learn how to find the greatest common factor (gcf) and the least common multiple (lcm) using liste of factors and multiples of two and three whole numbers.

Explorer all lessons of Arithmetic