Question

What is the fundamental theorem of arithmetic?Explain.​

Answer 1

Answer:

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The fundamental theorem of arithmetic (FTA), also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is prime itself or is the product of a unique combination of prime numbers.

Answer 2

THE FUNDAMENTAL THEOREM OF ARITHMETIC

We are familier with prime numbers like 2, 3, 5, 7,. 11, 13, 17, 19, 23, 29, 31, 37, ...

We also know that any natural number can be written as a product of its prime factors.

For example:

4 = 2 × 2 = 2²

18 = 2 × 3 × 3 = 2 × 3²

90 = 2 × 3 × 3 × 5 = 2 × 3² × 5

245 = 5 × 7²

260 = 2 × 2 × 5 × 13 = 2² × 5 × 13

21252 = 2 × 2 × 3 × 7 × 11 × 23 = 2² × 3 × 7 × 11 × 23 and so on.

FACTOR THREE

We are aware that any number cabe written as a product of powers as a product of powers of primes using the factor tree.

ILLUSTRATION : Express 1092 as product of its prime factors.

Solution : Refer to the attached picture.

OBSERVATION

The above observation lends us to a conjecture that every composite number can be written as the product of primes. In fact, this statement is true and is called the Fundamental Theorem of Arithmetic because pf its basic relevance (importance) in the development of number theory.

Now, let us formally state this theorem :

THEOREM (FUNDAMENTAL THEOREM OF ARITHMETIC)

Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

It is noteworthy that the prime factorization of a natural number is unique, except for the order of its factors.

General statement : In general, we factorise a given composite number 'a' as a = p₁p₂p₃.....pᵤ, where p₁p₂p₃.....pᵤ, are primes and written in ascending order, i.e., p₁ ≤ p₂ ≤ p₃ ≤ ..... ≤ pᵤ.