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of a single factor is the factor itself), the theorem is often stated as: every positive integer can be represented uniquely as a product of prime numbers, up to the order of the factors.

This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; for example,

The theorem generalizes to other algebraic structures that are called unique factorization domains

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