The Fundamental Theorem of Arithmetic states that,
"Every composite number can be expressed (factorised) as a product of the powers of primes, and this factorisation is unique, except for the order in which the prime factors occur."
Actually, it says a lot more.
This theorem states that any composite number can be written as a product of prime numbers in a "unique (one and only one)" way, except for the order in which the primes occur.
That is, if we don't care about the order of the primes, there is one and only one way to write any composite number as a product of primes.
Consider a composite number 210. Its prime factorisation is 2×3×5×7.
But we can also write this prime factorisation in the order 3×5×7×2.
That is, we consider 2×3×5×7 to be the same as 3×5×7×2, or any other possible order in which these primes could be written.
Hence, it is clear that the prime factorisation of 210 is unique except for the order of its factors.
Example -
Hence, the fundamental theorem of arithmetic is stated in the following way also:
"Except for the order of the factors, the prime factorisation of a natural number is unique."
Example-
Hence, once it is decided that the factors will be arranged ascendingly, then the way the number is factored, is unique.
This definition is called the Fundamental Theorem of Arithmetic because of its fundamental important role in the study of integers.
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