Integer Factorization

When we talked about factors and multiples we learned how to find factors of number.

This allows to write number as product of its factors.

Recall that factors of 12 are 1,2,3,4,6,12.

So, we can write that ${12}={1}\cdot{12}$ , ${12}={2}\cdot{6}$ and ${12}={3}\cdot{4}$ .

This means that there is more than one representation of same number as product of its factors.

We can find one more integer factorization of 12: since ${4}={2}\cdot{2}$ then ${12}={4}\cdot{3}={2}\cdot{2}\cdot{3}$ .

Example 1. Find all possible factorizations of 18.

Factors of 18 are 1,2,3,6,9,18.

So, ${18}={1}\cdot{18}={2}\cdot{9}={3}\cdot{6}$ .

Also, since 6 and 9 can also be factored as ${6}={2}\cdot{3}$ and ${9}={3}\cdot{3}$ then

${18}={3}\cdot{6}={3}\cdot{2}\cdot{3}$ and ${18}={2}\cdot{9}={2}\cdot{3}\cdot{3}$ . Last two factorizations are same (we just swaped factors), so 18 can be represented in the following way:

${18}={1}\cdot{18}={2}\cdot{9}={2}\cdot{6}={2}\cdot{3}\cdot{3}$ .

Factorization ${18}={2}\cdot{3}\cdot{3}$ can be rewritten using exponents as ${18}={2}\cdot{{3}}^{{2}}$ .

One more example.

Example 2. Find all possible factorizations of -15.

We first find factorization of 15.

Factors of 15 are 1,3,5,15.

So, ${15}={1}\cdot{15}={3}\cdot{5}$ .

Now, to find factorization of -15 place minus in front of each factorization: $-{15}=-{1}\cdot{15}=-{3}\cdot{5}$ .

Now, exercises.

Exercise 1. Find all possible factorizations of 20.

Answer: ${20}={1}\cdot{20}={2}\cdot{10}={4}\cdot{5}={{2}}^{{2}}\cdot{5}$ .

Another exercise.

Exercise 2. Find all possible factorizations of -45.

Answer: $-{45}=-{1}\cdot{45}=-{3}\cdot{15}=-{5}\cdot{9}=-{{3}}^{{2}}\cdot{5}$ .