The Fundamental Theorem of ArithmeticThe fundamental theorem of arithmetic states that every positive integer can be factored in one and only one way into a product of primes. For example, 60 can be expressed as 2^{2.}3^{.}5. No other combination of prime factors (excluding different orderings of 2, 2, 3, and 5) will yield 60. Note that 1 is not considered prime because its inclusion as a prime number would complicate the statement of many theorems including this one. For example, if 1 were prime, then 60 could equal both 2^{2.}3^{.}5 and 1^{.}2^{2.}3^{.}5. For a proof of the fundamental theorem of arithmetic, see Euclid's Algorithm and read the entire page.

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