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If you disregard different ordering
of the factors, any positive integer can be expressed as the product
of prime factors in exactly one way.
A form of the proof is attributed
to Euclid, approximately 325 BC. It is a direct consequence of Euclid's
"First Theorem", which establishes that if any prime number p divides
any product a X b, then either p divides a or p divides b. In other
words, it is not possible for a prime number to divide a product, without
also dividing at least one of the factors in the product.
The Fundamental Theorem of
Arithmetic is used repeatedly in the proofs of many elementary results
of number theory.
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