Fundamental Theorem of Arithmetic Definition

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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."

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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.

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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.

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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.

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Example -

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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."

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Example-

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Hence, once it is decided that the factors will be arranged ascendingly, then the way the number is factored, is unique.

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This definition is called the Fundamental Theorem of Arithmetic because of its fundamental important role in the study of integers.

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