Fundamental theorem of arithmetic

It can be time consuming to list all multiples until one is found in common. There is a more efficient way to find the least common multiple and greatest common factor. This method is based on the most important and basic idea about whole numbers The Fundamental Theorem of Arithmetic. 

THE FUNDAMENTAL THEOREM OF ARITHMETIC states that every whole number greater than 1 is the product of prime factors. Furthermore, these prime factors are unique, and there is exactly one set of prime factors. 

It is a good idea to know the primes less than 100, namely 2, 3, 5, 7, 11, 13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89, and 97. The fundamental theorem of arithmetic states that any positive integer can be represented as a product of primes in exacdy one way—not counting different ordering. The primes can, thus, be considered the atoms of integers. Euclid proved that there is no largest prime—they go on forever. The proof runs as follows. Assume, to the contrary, that there does exist a largest prime, say, p. Then, consider the number... 

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