1.1. The Fundamental Theorem. The positive integers are the integers 1, 2, 3, . . . . The prime numbers are those integers larger than 1 that can be factored into two positive integers in exactly one way (not paying attention to order). Thus 2, 3, 5, 7, 11, . . . are primes, whereas 1, 4, 6, 8, 9, 10, . . . are not primes. These non-prime integers > 1 are called composite numbers: to see that 10 is composite note that we can factor it in two distinct ways, as 1 × 10 and as 2 × 5. When one studies questions involving integers one quickly finds that it is useful to break integers down into their smallest component parts, that is to factor them into prime numbers. Thus 35 is 5 × 7, and 90 is 2 × 3 × 3 × 5, and so on; in fact, every positive integer can be factored in such a manner. A factorization into primes cannot be decomposed any further since none of the component primes can be factored again. From calculations it appears that there is only one way to factor a given integer, though this does not seem to be so easy to prove. However if true then it does give a solid foundation to any study of the positive integers, and so the result is considered to be the most fundamental in arithmetic:
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