High Quality Content by WIKIPEDIA articles! In number theory, the Fundamental Theorem of Arithmetic (or Unique-Prime-Factorization Theorem) states that any integer greater than 1 can be written as a unique product (up to ordering of the factors) of prime numbers. For example,are two examples of numbers satisfying the hypothesis of the theorem that can be written as the product of prime numbers. Intuitively, this theorem characterizes prime numbers uniquely in the sense that they are the "fundamental numbers. Proof of existence of a prime factorization is straightforward: proof of uniqueness is more challenging. Some proofs use the fact that if a prime number p divides the product of two natural numbers a and b, then p divides either a or b, a statement known as Euclid's lemma. Since multiplication on the integers is both commutative and associative, it does not matter in what way we write a number greater than 1 as the product of primes; it is generally common to write the (prime) factors in the order of smallest to largest.

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