81 lines
2.9 KiB
Markdown
81 lines
2.9 KiB
Markdown
|
---
|
||
|
|
tags:
|
||
|
|
- Mathematics
|
||
|
|
- Prealgebra
|
||
|
|
- factors
|
||
|
|
- divisors
|
||
|
|
---
|
||
|
|
|
||
|
|
## Factors and divisors
|
||
|
|
|
||
|
|
The terms **factor** and **divisor** are used interchangeably. They are different ways of expressing the same mathematical truth and this is because of the inverse relationship between division and multiplication.
|
||
|
|
|
||
|
|
### Divisors
|
||
|
|
|
||
|
|
>
|
||
|
|
> For a number $n$, its divisor is any number that divides $n$ evenly without remainder: $$ \frac{a}{b} = 0 $$
|
||
|
|
|
||
|
|
In this operation, $a$ is the **divisor**, $b$ is the **dividend** and $0$ is the **quotient**.
|
||
|
|
|
||
|
|
### Factors
|
||
|
|
|
||
|
|
>
|
||
|
|
> For a given number $n$, its factors are any pair of numbers that when multiplied together return $n$ as the product: $$ a \cdot b = n $$
|
||
|
|
|
||
|
|
We can see the relationship consists in the fact that factors are associated with multiplication and divisors are associated with division: two different perspectives on the same number relationships.
|
||
|
|
|
||
|
|
For example, 6 is both a factor and divisor of 18 and 24. To be precise, it is the greatest common divisor of these two numbers.
|
||
|
|
|
||
|
|
As a divisor:
|
||
|
|
$$
|
||
|
|
\\frac{18/6}{24/6} = \frac{3}{4}
|
||
|
|
$$
|
||
|
|
|
||
|
|
As a factor:
|
||
|
|
$$
|
||
|
|
\\frac{3 \cdot 6}{4 \cdot 6} = \frac{18}{24}
|
||
|
|
$$
|
||
|
|
|
||
|
|
When we divide by the common divisor is acts as a divisor. When we multiply by the common divisor it acts as a factor. The fact that the fractions are [equivalent](Equivalent%20fractions.md) in both cases indicates that the properties are equivalent.
|
||
|
|
|
||
|
|
## Greatest common divisor
|
||
|
|
|
||
|
|
>
|
||
|
|
> For two two integers $a, b$, $D$ is a common divisor of $a$ and $b$ if it is a divisor of both. The greatest common divisor is the largest value that $D$ can be whilst remaining a divisor to both $a$ and $b$.
|
||
|
|
|
||
|
|
### Demonstration
|
||
|
|
|
||
|
|
*Find the greatest common divisor of $18$ and $24$*
|
||
|
|
|
||
|
|
The divisors of 18:
|
||
|
|
$$1, 2, 3, 6, 9, 18$$
|
||
|
|
|
||
|
|
The divisors of 24:
|
||
|
|
$$ 1, 2, 3, 4, 6, 8, 12, 24$$
|
||
|
|
|
||
|
|
Thus the common divisors are:
|
||
|
|
$$ 1, 2, 3, 6 $$
|
||
|
|
|
||
|
|
The largest value in the above set is 6, thus 6 is the greatest common divisor.
|
||
|
|
|
||
|
|
## Heuristics for finding divisors
|
||
|
|
|
||
|
|
1. For dividend $n$ , if $n$ ends in an even number or zero, $n$ is **divisible by 2**.
|
||
|
|
1. $\frac{12}{2} = 6$
|
||
|
|
1. $\frac{84}{2} = 42$
|
||
|
|
1. For dividend $n$ if the sum of the digits is divisible by 3 then $n$ is **divisible by 3**.
|
||
|
|
1. $\frac{72}{3} = 24$
|
||
|
|
1. $\frac{21}{3} = 7$
|
||
|
|
1. For a dividend $n$ if the number represented of the last two digits of $n$ divides by 4 then $n$ is divisible by 4
|
||
|
|
1. $\frac{324}{4} = 81$
|
||
|
|
1. $\frac{532}{4} = 133$
|
||
|
|
1. For a dividend $n$, if the last digit of $n$ is divisible by 0 or 5, then $n$ is divisible by 5.
|
||
|
|
1. $\frac{25}{5} = 5$
|
||
|
|
1. For a dividend $n$, if $n$ is divisible by 2 and 3, then $n$ is divisible by 6.
|
||
|
|
1. $\frac{12}{6} = 2$
|
||
|
|
1. $\frac{18}{6} = 3$
|
||
|
|
1. For a dividend $n$, if the last three digits of $n$ are divisible by 8, then $n$ is divisible by 8.
|
||
|
|
1. $\frac{73024}{8} = 9128$
|
||
|
|
1. For a dividend $n$, if the sum of the digits of $n$ is divisible by 9 then $n$ is divisible by 9.
|
||
|
|
1. $\frac{117}{9} = 13$
|