# 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.

Ready for exercises?

**Exercise 1.** Find all factors of 1.

**Answer**: -1 and 1.

Next one.

**Exercise 2.** Find all factors of 24.

**Answer**: -24,-12,-8,-6,-4,-3,-2,-1,1,2,3,4,6,8,12,24.

Factors of negative numbers are done in same way.

**Exercise 3.** Find all factors of -25.

**Answer**: -5,-1,1,5.

Now, exercise in finding multiples.

**Exercise 4.** Find all multiples of 7.

**Answer**: ...,-28,-21,-14,-7,0,7,14,21,28,...

What about multiples of negative numbers? Same technique.

**Exercise 5.** Find all multiples of -4.

**Answer**: ...,-16,-12,-8,-4,0,4,8,12,16,...