Fundamental Theorem of Arithmetic

Let us take 36 and factorize it. It can be factorised in many ways.

First let us divide by 2 as many times as possible and the divide by 3 as many times as possible. We would get 36 = 2 X 2 X 3 X 3. It can be written as<a name="_Hlk41901907">.</a>

Or we can follow another way, divide by 6 which gives 36 = 6 X 6. Each of the 6s can further be written as 2X3. Hence we get 36 = 2 X 3 X 2 X 3. This can also be written as .

Whichever way we factorize 36 we get a unique solution that 36 = .

Hence, if we consider primes to be the alphabets of math, then every number can be considered as a word having a unique spelling! Hence primes are also considered the building blocks of numbers.

Fundamental Theorem of Arithmetic

The idea of unique prime factorization was discovered first as part of number theory explorations. It was then called the unique factorization theorem or the unique-prime-factorization theorem.

This was considered so important that it was later called the Fundamental Theorem of Arithmetic. It can be formally stated as given below.

"Every Composite Number can be expressed or factorized as the product of prime numbers and the factorization is unique for every composite number except the order of prime numbers"

Not Obvious!

We need to be clear that this theorem of unique factorization is not very obvious. For example, it is not obvious that 23X 1759 is not equal to 53 X 769, given that each of the four numbers is prime. The totals are very close differing in only one place! Only actual multiplication (23X1759=40,457 & 53X 769=40,757) will show that they are not equal.

Euclid’s theorem that there are an infinite number of primes, uses the above idea that any number can be represented as a product of powers of primes in only one way! This fact of unique factorization was observed by many mathematicians over many centuries. A formal proof was worked by <a href="https://www.britannica.com/biography/Carl-Friedrich-Gauss">Carl Friedrich Gauss</a> in 1801.

The Curious Cases of 1 & 2

1 is considered not a prime number whereas 2 is considered a prime number. These are favourite examination questions. But the logic of above definitions is not clear even to teachers. The logic lies in the Fundamental Theorem of Arithmetic.

Why 1 is Not a Prime?

As per the theorem, 6 can be written as 2X3. If 1 was considered a prime number, then 6 can be written as 1X2X3. But since the value of any power of 1 is always one 6 can also be written as or . So the property of uniqueness is lost!

This is the reason 1 has been defined as a non-prime number. 1 which is a factor of any number will not be considered a prime factor of that number!

Why 2 is Prime?

2 is considered a prime number for two reasons.

1. All whole numbers are evenly divided between odd & even numbers. All even numbers will have 2 or a power of 2 as a factor. Hence 2 will appear as a factor of most numbers. Hence 2 should be considered as a prime number.

2. A prime number is one which only has two factors, 1 and the number itself. The factors of 2 are 1 & 2 itself. Hence 2 satisfies the criteria for a prime number.

The confusion has arisen because all primes are expected to be odd numbers, as every even number can be divided by 2.

We now see that 2 is a prime number. It is the only “even” prime number! All other primes are necessarily odd numbers.