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 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=2×693 , and we proceed from there.

factor tree of 1386

This shows that the prime factorization of 1386 is 2×3×3×7×11 .

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 .

## Answers ( )

Answer: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 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=2×693 , and we proceed from there.

factor tree of 1386

This shows that the prime factorization of 1386 is 2×3×3×7×11 .

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 .