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Math Help - Euclid's lemma and induction

  1. #1
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    Euclid's lemma and induction

    Use Euclid's Lemma and induction to prove that if p and
    q1,q2,q3,...,qn (1,2,3 and n are subscripts) are prime numbers
    and p divides q1q2q3...qn ( the product), then p=qi for some
    i=1,2,3....n.
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  2. #2
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    Quote Originally Posted by JCIR View Post
    Use Euclid's Lemma and induction to prove that if p and
    q1,q2,q3,...,qn (1,2,3 and n are subscripts) are prime numbers
    and p divides q1q2q3...qn ( the product), then p=qi for some
    i=1,2,3....n.
    Euclid's lemma tells you that this is true for n=2.

    Now suppose it true for some n=k, then we have for any prime numbers:

    p, q_1, .., q_{k+1}

    Then by Euclids lemma if p | q_1q_2...q_{k+1} then either p|q_1q_2...q_{k} or p|q_{k+1}. But we already have by assumption that if p|q_1q_2...q_{k} then p divides one of p_1, ...,p_k. Hence the result if true for n=k is true for n=k+1, and with the base case we have a proof by induction for all integers n \in \{2,..\}.

    RonL
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