Unique Prime Factorization
TheFundamental Theorem of Arithmeticstates that every natural number greater thancan be written as a product ofprime numbers, and that up to rearrangement of the factors, this product isunique. This is called theprime factorizationof the number.
Example:
can be written as, or, or, or. But there is only one way to write it as a product where all the factors are primes:
This is the prime factorization of, often written with exponents:
For a prime number such asor, the prime factorization is simply itself. Anycompositenumber (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因素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. We start by noticing thatis even, sois a factor. Dividing by, we get, and we proceed from there.
This shows that the prime factorization ofis.
You can use prime factorizations to figure outGCFs (Greatest Common Factors),LCMs (Least Common Multiples), and the number (and sum) of divisors of.