# Unique Prime Factorization

The
**
Fundamental
Theorem of Arithmetic
**
states that every natural number greater than
$1$
can be written as a product of
prime numbers
, and that up
to rearrangement of the factors, this product is
**
unique
**
. This is
called the
**
prime
factorization
**
of the number.

**
Example:
**

$36$ can be written as $6\times 6$ , or $4\times 9$ , or $3\times 12$ , or $2\times 18$ . But there is only one way to write it as a product where all the factors are primes:

$36=2\times 2\times 3\times 3$

This is the prime factorization of $36$ , often written with exponents:

$36={2}^{2}\times {3}^{2}$

For a prime number such as
$13$
or
$11$
, the prime factorization is simply itself. Any
**
composite
**
number (that is, a whole number with more than two factors) has a non-trivial prime factorization.

The prime factorization of a number can be found using a
**
factor tree
**
. Start by finding two factors which, multiplied together, give the number. Keep splitting each branch of the tree into a pair of factors until all the branches terminate in prime numbers.

Here is a factor tree for $1386$ . We start by noticing that $1386$ is even, so $2$ is a factor. Dividing by $2$ , we get $1386=2\times 693$ , and we proceed from there.

This shows that the prime factorization of $1386$ is $2\times 3\times 3\times 7\times 11$ .

You can use prime factorizations to figure out GCF s (Greatest Common Factors), LCM s (Least Common Multiples), and the number (and sum) of divisors of $n$ .

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