The Fundamental Theorem of Arithmetic

The 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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