Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that for every integer \color{red}n more than 1, {\color{red}n}>1, is either a prime number itself or a composite number which can be expressed in only one way as the product of a unique combination of prime numbers.

Each prime factor occurs in the same amount regardless of the order of the product of the prime factors. In other words, the number of each prime factor of an integer is always constant.

That is why in the case of a repeated prime number in the prime factorization, we will just write one copy of the prime number with a corresponding exponent that equals to how many times it shows up in the factorization.

Furthermore, the best way to demonstrate the uniqueness of the prime factorization of an integer is to write the prime factors in ascending order, when multiplying them to get the product. By doing so, the uniqueness of prime factorization is accentuated and emphasized.

Therefore, no integers greater than 1 have the same prime factorization. This means every integer greater than 1 has a unique prime factorization.


Breakdown of the Description of the Fundamental Theorem of Arithmetic

▶︎ A prime number is a positive integer larger than \color{red}1 with exactly two factors: namely, 1 and itself. The integer 1 is neither prime nor composite. It is not prime since it has exactly one factor. In the same manner, integer 1 is not composite because a composite number must have three or more factors.

▶︎ Below is the set of the integers. 

the set of integers. its subsets are the whole numbers and natural numbers.

What we want though are just the integers greater than 1. This means the numbers are

\large{2,3,4,5,6,7,8,9,10,…}

You may have observed also that the integers greater than 1 is the same as the set of natural numbers excluding 1.

▶︎ Remember that rearranging the order of the factors of an integer does not make it unique. 

For example, the integer 360 can be prime factorized in different ways. But I will only show four variations below to drive the point.

four different ways how the integer 360 is prime factorized

The key point here is that the order in which we multiply the prime factors of the integer 360 may vary but the number of each unique prime number is constant. That is, for the integer 360, we will always have three 2’s, two 3’s, and one 5.

Although the orders of the prime factors are different, they are considered to be the same if we disregard the order of writing the prime factors. Thus, when talking about unique prime factorization, we must ignore the order.

▶︎ To accentuate the uniqueness of the prime factorization of an integer greater than 1, we write the factors which are prime numbers in increasing order.

In addition, in the case of repeated prime numbers, we can write them in compact form using exponents.

I’m currently working on this lesson. This is not finished yet. Thank you for your patience!