# How do you do Prime Factorization

## Related calculator: Prime Factorization Calculator

When we talked about integer factorization we noticed that integer number can have more than one factorization.

For example, `18=1*18=2*9=3*6=2*3^2`.

However, if we use only prime numbers as factors than such factorization is unique (ignoring order of factors).

**Fact.** Every composite number can be uniquely represented as a product of prime factors.

For example, `18=2^2*3`. We can write `18=3*2^2`, but we only swap factors, so this representation is the same.

To find **prime factorization of a number** we start by dividing number by the smallest prime number (2), then we move to the next prime number until we get prime number in result.

**Example 1.** Find prime factorization of 60.

Try to divide 60 by 2: `60=2*30`.

Try to divide 30 by 2: `30=2*15`.

Try to divide 15 by 2: not divisible.

Move to the next prime number 3.

Try to divide 15 by 3: `15=3*5`.

We are done, because 5 is prime number.

So, `60=2*2*3*5=2^2*3*5`.

Next example.

**Example 2.** Find prime factorization of 19980.

Divide 19980 by 2: `19980=2*9990`.

Divide 9990 by 2: `9990=2*4995`.

Divide 4995 by 2: not divisible.

Try next prime number 3.

Divide 4995 by 3: `4995=3*1665`.

Divide 1665 by 3: `1665=3*555`.

Divide 555 by 3: `555=3*185`.

Divide 185 by 3: not divisible.

Try next prime number 5.

Divide 185 by 5: `185=5*37`.

We are done since 37 is prime number.

So, `19980=2*2*3*3*3*5*37=2^2*3^3*5*37`.

Last example.

**Example 3.** Find prime factorization of 625.

Divide 625 by 2: not divisible.

Divide 625 by 3: not divisible.

Divide 625 by 5: `625=5*125`.

Divide 125 by 5: `125=5*25`.

Divide 25 by 5: `25=5*5`.

We are done since 5 is prime number.

So, `625=5*5*5*5=5^4`.

Now, it is time to practice.

**Exercise 1.** Find prime factorization of 990.

**Answer**: `2*3^2*5*11`.

Slightly harder exercise.

**Exercise 2.** Find prime factorization of 1725.

**Answer**: `3*5^2*23`.

And final exercise.

**Exercise 3.** Find prime factorization of 343.

**Answer**: `7^3`.