Question

State the fundamental theorem of arithmetic

Answer 1

Hint: By definition
Prime number: -A number that has only two factors i.e. \[1\] and itself is a prime number.
Composite number: - When a number can be divided up exactly and has more than 2 factors it is called a composite number

Complete step-by- step solution:
Definition of fundamental theorem of arithmetic: - Every composite number can be expressed as a product of prime numbers and the representation is unique except for the order in which the prime factor occurs.
Example: -\[75\]
Prime factors of 75 are:
\[ = 1 \times 3 \times 5 \times 5\]
\[ = 1 \times 3 \times (5 \times 5)\]
\[ = 1 \times 3 \times {5^2}\]
Now arranging them in ascending order, we get:
\[ = 1 \times 3 \times {5^2}\]
Now arranging them in descending order, we get:
\[ = {5^2} \times 3 \times 1\]
We can easily see that the composite number 75 is expressed as a product of prime numbers and the representation is unique but we have expressed it in different order.

Note: Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number or can be expressed in the form of primes. In other words, all the natural numbers can be expressed in the form of the product of its prime factors. To recall, prime factors are the numbers which are divisible by 1 and itself only every natural number can be expressed in the form of the product of the power of its primes. This statement is known as the Fundamental Theorem of Arithmetic, unique factorization theorem or the unique-prime-factorization theorem.
In the fundamental theorem of arithmetic c-very composite number expressed is a product of prime numbers in ascending and descending order.