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 × 6, or 4 × 9, or 3 × 12, or 2 × 18. But there is only one way to write it as a product where all the factors are primes:
36 = 2 × 2 × 3 × 3
This is the prime factorization of 36, often written with exponents:
36 = 22 × 32
For a prime number such as 13 or 101, the prime factorization is simply itself.
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 × 693, and we proceed from there.
This shows that the prime factorization of 1386 is 2 × 3 × 3 × 7 × 11.
You can use prime factorizations to figure out GCFs (Greatest Common Factors), LCMs (Least Common Multiples), and the number (and sum) of divisors of n.
