Textbook Problem

a. Use mathematical induction and Euclid’s lemma to prove that for every positive integer s. if p and $q_1,q_2,\dots ,q_s$ are prime numbers and $p|q_1{q}_2\cdots q_s$ , then $p=q_i$ for some i with $1\le i\le s$.

b. The uniqueness part of the unique factorization theorem for the integers says that given any integer n if $n=p_1{p}_2\cdots p_r\equiv q_1{q}_2\cdots q_s$ for some positive integers r and s and prime numers $p_1\le p_2\le \cdots \le p_r$and $q_1\le q_2\le \cdots \le q_s$ then $r=s$ and $p_i=q_i$ for every integer i with $1\le i\le r$ .

Use the result of part (a) to fill in the details of the following sketch of a proof: Suppose than n is an integer with two different prime faclorizations: $n=p_1{p}_2\cdots p_t=q_1{q}_2\cdots q_u$ . All the prime factors that appear on both sides can be cancelled (as many times as they appear on both sides) to arrive at the situation where $p_1{p}_2\cdots p_r=q_1{q}_2\cdots q_s,\text{}{p}_1\le p_2\le \cdots \le p_r,q_1\le q_2\le \cdots \le q_s$ , and $p_i\ne q_j$ for any integers i and j. Then use part (a) to deduce a contradiction. and conclude that the prime factorization of n is unique except, possibly, for the order in which the prime factors are written.