# How to Calculate Factorials,Multiples and Factors of Numbers

## Factorials

The factorial of a number which is positive in nature is a multiplication sum.

How to calculate factorials involves multiplying the relevant positive number with a list of descending smaller numbers down to the number 1.

The symbol used to represent a factorial is an exclamation mark.  !

Thus writing down four factorial would be   4!.

With it representing the following:

4!  =  4 × 3 × 2 × 1  =  24

So in general, the factorial of a positive number  r  is:

r!  =   r  ×  ( r 1 )  ×  ( r 2 )  ×  ………  ×  1

### How to Calculate Factorials, 0!

A special case to mention when thinking about how to calculate factorials, is zero factorial,  0!.

As unexpectedly, the value of zero factorial is  1.

To get a picture of why this is the case, we can rewrite the general factorial rule above slightly differently.

r!  =   r  ×  ( r 1 )!

Now it’s the case that   1!  =  1.

So we’d have:    1!  =  1 × ( 1 1 )!.

Which is:    1!  =  1 × ( 0 )!.

Meaning that   0!   has to be equal to one for this to be true.

## Multiple of a Number

In Math, a multiple of a number is the result of multiplying a given number by an integer.

We can look at the number  6  as an example.

6 × 2 = 12   ,   6 × 1 = 6   ,   6 × 3 = 18

6 × -1 = -6   ,   6 × -4 = -24

So  12 , 6 , 18 , -6  and  -24  are all multiples of the number  6.

They are all the result of multiplying 6 by an integer.

It’s the case that fractions and irrational numbers can have multiples also.

\bf{\frac{1}{4}} × 1 = \bf{\frac{1}{4}}   ,   \bf{\frac{1}{4}} × 3 = \bf{\frac{3}{4}}

are sums that give multiples of  \bf{\frac{1}{4}}.

An irrational number such as π, will have multiples of the form:

2π    5π    etc.

## Factor of a Number

A factor of a certain number, is another number that can divide the certain number evenly, with no remainder.
This also includes the certain number itself.

Take the number  16  for example.

The numbers   1 , 2 , 4 , 8 , 16   are factors of   16.

16 ÷ 1 = 16  ,  16 ÷ 2 = 8 ,  16 ÷ 4 = 4 ,  16 ÷ 8 = 2 ,  16 ÷ 16 = 1  )

### Factors and Factor Pairs:

A ‘factor pair’ is two factors of a given number, that multiply together to result in that given number.

The factor pairs of  16  are:

1  and  16   ,   2  and  8.

=>    (  1 × 16 = 16   ,   2 × 8 = 16  )

Similarly, the factor pairs of  12  are:

1  and  12   ,   2  and  6   ,   3  and  4.

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