Question

1.26. Let {p1, p2, ..., pr} be a set of prime numbers, and let N = p1p2 ... pr + 1. Prove that N is divisible by some prime not in the original set. Use this fact to deduce that there must be infinitely many prime numbers. (This proof of the infinitude of primes appears in Euclid's Elements. Prime numbers have been studied for thousands of years.) 1.27. Without using the fact that every integer has a unique factorization into primes, prove that if gcd(a, b) = 1 and if a | bc, then a | c. (Hint. Use the fact that it is possible to find a solution to au + bv = 1.) 1.28. Compute the following ord_p values: (a) ord_2(2816). (b) ord_7(2222574487). (c) ord_p(46375) for each of p = 3, 5, 7, and 11.

          1.26. Let {p1, p2, ..., pr} be a set of prime numbers, and let
N = p1p2 ... pr + 1.
Prove that N is divisible by some prime not in the original set. Use this fact to deduce that there must be infinitely many prime numbers. (This proof of the infinitude of primes appears in Euclid's Elements. Prime numbers have been studied for thousands of years.)

1.27. Without using the fact that every integer has a unique factorization into primes, prove that if gcd(a, b) = 1 and if a | bc, then a | c. (Hint. Use the fact that it is possible to find a solution to au + bv = 1.)

1.28. Compute the following ord_p values:
(a) ord_2(2816).
(b) ord_7(2222574487).
(c) ord_p(46375) for each of p = 3, 5, 7, and 11.

1.26. Let p1, p2, ..., pr be a set of prime numbers, and let
N = p1p2 ... pr + 1.
Prove that N is divisible by some prime not in the original set. Use this fact to deduce that there must be infinitely many prime numbers. (This proof of the infinitude of primes appears in Euclid's Elements. Prime numbers have been studied for thousands of years.)

1.27. Without using the fact that every integer has a unique factorization into primes, prove that if gcd(a, b) = 1 and if a | bc, then a | c. (Hint. Use the fact that it is possible to find a solution to au + bv = 1.)

1.28. Compute the following ordp values:
(a) ord2(2816).
(b) ord7(2222574487).
(c) ordp(46375) for each of p = 3, 5, 7, and 11.

Added by Logan W.

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