Unformatted text preview: 1.6. DIVISIBILITY IN THE INTEGERS 27 Definition 1.6.3. A natural number is prime if it is greater than 1 and not divisible by any natural number other than 1 and itself. To show formally that every natural number is a product of prime num bers, we have to use mathematical induction. Proposition 1.6.4. Any natural number other than 1 can be written as a product of prime numbers. Proof. We have to show that for all natural numbers n 2 , n can be written as a product of prime numbers. We prove this statement using the second form of mathematical induction. (See Appendix C.) The natural number 2 is a prime, so it is a product of primes (with only one factor). Now suppose that n > 2 and that for all natural numbers r satisfying 2 r < n , the number r is a product of primes. If n happens to be prime, then it is a product of primes. Otherwise, n can be written as a product n D ab , where 1 < a < n and 1 < b < n (by the definition of prime number). According to the induction hypothesis, each ofnumber)....
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 Fall '08
 EVERAGE
 Algebra, Integers, Prime Numbers, Natural number, Prime number, Euclid

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