# What are Factors and Multiples

Let's see what are factors and multiples.

When we talked about divisibility of integers, we said that number ${a}$ is divisible ${b}$ if ${c}=\frac{{a}}{{b}}$ is integer number.

If ${a}$ is divisible by ${b}$, then ${b}$ is a factor of ${a}$ and ${a}$ is multiple of ${b}$.

For example, since $\frac{{12}}{{3}}={4}$ then 3 is a factor of 12 and 12 is multiple of 3.

Also, since $\frac{{12}}{{4}}={3}$ then 4 is a factor of 12 and 12 is multiple of 4.

It is known that division is inverse of multiplication, so we can write that ${12}={3}\cdot{4}$. Here, we clearly see that 3 and 4 are factors of 12.

But 12 has other factors as well. Since ${12}={1}\cdot{12}$ and ${12}={2}\cdot{6}$ then 1, 2, 6, 12 are also factors of 12.

Nice facts.

• Any number ${a}$ (except 1) always has at least two factors: 1 and ${a}$.
• ${a}$ is multiple of ${a}$ and 1.

Indeed, since ${a}={1}\cdot{a}$ then ${a}$ has two factors: 1 and ${a}$.

Also, ${a}$ is multiple of ${a}$ and 1.

Example 1. Find all positive factors of 8.

We need to check all numbers that are less or equal 8.

1 and 8 are factors of 8.

2 is factor of 8 because $\frac{{8}}{{2}}={4}$ (so 8 is divisible by 2).

4 is also factor of 8 because ${8}={2}\cdot{4}$.

3 is not factor of 8 because 8 is not divisible by 3.

5,6,7 are also not factors of 8.

So, positive factors of 8 are 1,2,4,8.

Next example.

Example 2. Find all factors of 10.

We need to check all numbers that are less or equal 10.

1 and 10 are factors of 10.

Other factors are 2 and 5.

So, positive factors are 1,2,5,10.

Now, we need to find negative factors. For this just write minus in front of positive factors: -1,-2,-5,-10.

So, factors of 10 are -10,-5,-2,-1,1,2,5,10.

Now, let's see how to find multiples.

Example 3. Find positive multiples of 5.

We start from 5, because 5 is multiple of 5.

Now we add 5: 5+5=10. This is another multiple.

Now add 5: 10+5=15. This is another multiple.

In a similar fashion we can find that positive multiples are 5,10,15,20,...

Ellipsis means that there are infinitely many multiples.

Next example.

Example 4. Find all multiples of 8.

First we find positive multiples.

8 is first multiple, 8+8=16 is second, 16+8=24 is third etc.

To find non-positive multiples we subtract 8.

8-8=0, 0-8=-8, -8-8=-16, -16-8=-24 etc.

So, multiples of 8 are ...,-24,-16,-8,0,8,16,24,...

To find multiple of number ${a}$ we need multiple it by all integers number.

For example, ${8}\cdot{\left(-{3}\right)}=-{24}$, ${8}\cdot{\left(-{2}\right)}=-{16}$, ${8}\cdot{0}={0}$, ${8}\cdot{1}={8}$, ${8}\cdot{2}={16}$ etc.

Exercise 1. Find all factors of 1.

Next one.

Exercise 2. Find all factors of 24.

Factors of negative numbers are done in same way.

Exercise 3. Find all factors of -25.