Question

What is fundamental theorem of arithmetic?

Answer 1

Hint: The fundamental theorem of arithmetic deals with how each number can be expressed as either a prime number or as a product of prime numbers. State the theorem, explain each of the concepts involved, and give examples.

Complete step by step answer:
Let us first state the fundamental theorem of arithmetics. It states that any integer greater than 1 is a prime number or can be written as a unique product of prime numbers.
Now, we know that integers include the set of negative, positive integers, and zero. But in the fundamental theorem of arithmetic, only the positive integers are considered, that too after the integer 1. So, that includes 2, 3, 4, 5, ….. infinity.
Next, it is mentioned that the integers greater than 1, i.e, 2, 3, 4, 5, … can be expressed as prime numbers or as a product of prime numbers. So, let us understand what a prime number is.
A prime number is one which has only two factors, that are 1 and itself. It is not exactly divisible by any other numbers. So, we have examples like 2, 3, 5, 7, …. Here 2 is the smallest prime number and the only even prime number too.
Let us write the list of integers that are not prime, i.e are composite. We have 4, 6, 8, 9, 10, …. As per the theorem, we must be able to express them as a unique product of prime numbers. So, let us take 4 as an example. We know that it can be expressed as $ 4=2\times 2 $ . Similarly, some other examples are as below,
 $ \begin{align}
  & 10=2\times 5 \\
 & 15=3\times 5 \\
 & 24=2\times 2\times 2\times 3 \\
\end{align} $

Note:
 We can also use the prime factorization technique which expresses any integer greater than 1 as a product of its prime factors. The theorem can be applied to find the greatest common divisor, least common multiples of integers. The order of the prime numbers in the product is not considered, i.e we can express $ 10=2\times 5 $ as well as $ 10=5\times 2 $ .