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In [[number theory]], the '''fundamental theorem of arithmetic''', also called the '''unique factorization theorem''' or the '''unique-prime-factorization theorem''', states that every [[integer]] greater than 1<ref>Under the [[empty product|empty product rule]] the theorem reduces to: every positive integer has unique prime factorization.</ref> either is prime itself or is the product of [[prime number]]s, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.<ref>{{harvtxt|Long|1972|p=44}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=53}}</ref><ref>{{Harvtxt|Hardy|Wright|2008|loc=Thm 2}}</ref> For example,
1200 = 2{{sup|4}} × 3{{sup|1}} × 5{{sup|2}} = 3 × 2× 2× 2× 2 × 5 × 5 = 5 × 2× 3× 2× 5 × 2 × 2 = etc.
The theorem is stating two things: first, that 1200 ''can'' be represented as a product of primes, and second, no matter how this is done, there will always be four 2s, one 3, two 5s, and no other primes in the product.
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