---
tags:
  - prealgebra
---

## 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. $\frac{117}{9} = 13$
