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Math Help - contradiction

  1. #1
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    contradiction

    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
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  2. #2
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    Quote Originally Posted by qwerty321 View Post
    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.
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