## 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:

2

*π*5

*π*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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