possible In Euclid's proof that there is no largest prime, a number can be formed by taking the product of primes and adding 1. The number 30,031 can be formed in this manner. Is 30,031 prime or composite? If it is composite, give its prime factorization. Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. 30,031 is composite. Its prime factorization is B. 30.031 is prime.
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Euclid's proof for the infinitude of primes states that if you take any finite list of prime numbers and multiply them together, then add 1 to the product, the resulting number cannot be divisible by any of the primes in your list. This new number is either prime Show more…
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Theorem 2.3 (Infinitude of primes): There are infinitely many prime numbers. PROOF: Suppose you have a finite list of prime numbers. Multiply all the prime numbers in your list together and call the result N. So, N is a positive integer. As N + 1 is again a positive integer bigger than 1, N + 1 has a prime factor p. Remember, you began with a list of prime numbers! If p were in your list, then p divides N (since N was the product of primes in your list) and p divides N + 1 (we chose p to be a factor of N + 1). By the two-out-of-three principle, p divides (N + 1) - N, so p divides 1. But no prime number is a factor of 1! Hence, the hypothesis ("if p were in the list") cannot be true. The prime number p is a "new prime," not in the list. Since no finite list of prime numbers contains all the prime numbers, there are infinitely many prime numbers.
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