hi..how can i prove by contradiction that
that ifp is a prime number and p divides a1a2...anwhere ai is an integer for i = 1; 2; 3... n, then p divides jai for some integer i
thank you
Assume by contradiction. Then $\displaystyle p\not | a_i$. However, $\displaystyle p|[a_1(a_2...a_n)]$. Since $\displaystyle p\not |a_1 \implies \gcd(a_1,p)=1$ and therefore $\displaystyle p|(a_2...a_n)$. Similar reasoning shows that $\displaystyle \gcd(a_2,p)=1$ and so $\displaystyle p|(a_3....a_n)$. Continue down this path until you reach $\displaystyle p|(a_{n-1}a_n)$ but then $\displaystyle p|a_n$, however, $\displaystyle p\not | a_n$. This is a contradiction.