hi..how can i prove by contradiction that
that if
p is a prime number and p divides a1a2...an
where ai is an integer for i = 1; 2; 3... n, then p divides jai for some integer i

thank you

2. Originally Posted by qwerty321
hi..how can i prove by contradiction that
that if
p is a prime number and p divides a1a2...an
where ai is an integer for i = 1; 2; 3... n, then p divides jai for some integer i

thank you
Assume by contradiction. Then $p\not | a_i$. However, $p|[a_1(a_2...a_n)]$. Since $p\not |a_1 \implies \gcd(a_1,p)=1$ and therefore $p|(a_2...a_n)$. Similar reasoning shows that $\gcd(a_2,p)=1$ and so $p|(a_3....a_n)$. Continue down this path until you reach $p|(a_{n-1}a_n)$ but then $p|a_n$, however, $p\not | a_n$. This is a contradiction.