In some situations, we are interested in the multiples and divisors of integers. The study of multiples and divisors makes it possible to understand the factorization of numbers as well as the concepts of the GCF (Greatest Common Factor) and LCM (Least Common Multiple).
A multiple of a number is the product of multiplying the number with some other integer.
The set of multiples of a number is the result of multiplying that number by each of the integers (|\mathbb{Z}|).
|12| is a multiple of |3|, because |3\times 4=12|.
The set of multiples of |3| is obtained by multiplying |3| by each of the elements in |\mathbb{Z}|.
||\left\{ \dots,\text{-}12,\text{-}9,\text{-}6,\text{-}3,0,3,6,9,12,\dots \right\}||
In general, the strictly positive multiples of a number are the main interest; in other words, the multiples of a number in |\mathbb{N}^*|.
For the multiples of the number |3|, for example, consider the following set. ||\left\{3,6,9,12,\dots \right\}||
A divisor of a number is an integer that divides the number without leaving a remainder. In other words, an integer is a divisor of another number if the quotient is also an integer.
The set of divisors of a number consists of all the integers that divide the number without leaving a remainder.
|4| is a divisor of |24| because |24\div 4=6|. |5| is not a divisor of |24| because |24\div 5=\color{red}{4.8}| (The quotient is not an integer).
The set of divisors of |24| is given by: ||\left\{\text{-}24,\text{-}12,\text{-}8,\text{-}6,\text{-}4,\text{-}3,\text{-}2,\text{-}1,1,2,3,4,6,8,12,24\right\}||
In general, the positive divisors of a number are the main interest; in other words, the divisors of a number in |\mathbb{N}|.
For the divisors of the number |24|, for example, we consider the following set: ||\left\{1,2,3,4,6,8,12,24\right\}||
There are several ways to list the divisors of a number. The easiest way is to consider all the possible divisors in ascending order.
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Consider all the possible divisors in ascending order.
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List all the divisors inside a pair of curly brackets.
While looking for the divisors of a number, note that if two consecutive divisors multiply together to give that number, then all that remains is to find the pairs of divisors to complete the list.
Give the set of divisors for |32|.
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Consider all the possible divisors in ascending order. ||\begin{align}\text{Does }1\text{ divide }32\text{ ?}&\Rightarrow \text{Yes}\\
\text{Does }2\text{ divide evenly into }32\text{ ?}&\Rightarrow\text{Yes}\\
\text{Does }3\text{ divide evenly into }32\text{ ?}&\Rightarrow\color{red}{\text{No}}\\
\text{Does }4\text{ divide evenly into }32\text{ ?}&\Rightarrow\text{Yes}\\
\text{Does }5\text{ divide evenly into }32\text{ ?}&\Rightarrow\color{red}{\text{No}}\\
\text{Does }6\text{ divide evenly into }32\text{ ?}&\Rightarrow\color{red}{\text{No}}\\
\text{Does }7\text{ divide evenly into }32\text{ ?}&\Rightarrow\color{red}{\text{No}}\\
\text{Does }8\text{ divide evenly into }32\text{ ?}&\Rightarrow\text{Yes}\\ \dots \end{align}||
Note that the last two consecutive factors in this list, |4| and |8|, multiply together to give |32|.
At this step, the following divisors are listed: |\left\{\color{orange}{1},\color{blue}{2},4,8\right\}|
Using the trick given above, complete the pairs of divisors to finish the list.
So, ||\begin{align}4\times 8 &= 32\\
\color{blue}{2}\times \color{blue}{16}&=32\\
\color{orange}{1}\times \color{orange}{32}&=32\end{align}|| -
List all the divisors inside a pair of curly brackets.
The set of divisors of |32| is therefore |\left\{\color{orange}{1},\color{blue}{2},4,8,\color{blue}{16},\color{orange}{32}\right\}|.
To quickly find the divisors of a given number, it is useful to use the divisibility rules.