MATH135 Study Guide - Quiz Guide: Coprime Integers, Integer Factorization, Prime Factor

• School: University of Waterloo
• Department: Mathematics
• Course: MATH135
• Professor: Sean Mc Master

Document Summary

This material is from chapter 50 of the course notes. We shall prove that the prime factorization of a natural number is essentially unique. Recall that a prime number is an integer p 1 for which the only positive divisors are 1 and p itself. Also recall that a composite number is an integer n 1 which is not prime, i. e. , n = kl for some integers k 1 and l 1. The following two theorems have been proved before the midterm. Any integer n 1 is a product of nitely many primes. Then either n is prime, or there exists a prime factor p of n for which p ?n. By proposition 2, n is a product of more than one primes. Let p be the smallest prime factor of n. the quotient n/p 1 is a product of primes. From the choice of p, it follows that p n/p.