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# Unique Factorization Theorem

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The process of writing any positive integer as the product of its prime factors is probably familiar; it may be taken as self-evident that this can be done in only one way. Known as the Unique Factorization Theorem, this result of elementary number theory can be proved from basic axioms about the integers.

Theorem:

Any positive integer (≠1) can be expressed as a product of primes. This expression is unique except for the order in which the primes occur.

Thus, any positive integer n(≠1) can be written as where p1, p2,…, pr are primes, satisfying p1 < p2 <⋯< pr, and α1, α2,…, αr are positive integers. This is the prime decomposition of n. For example, writing 360=23×32×5 shows the prime decomposition of 360.

Subjects: Mathematics.

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