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Math Help - Another relatively prime problem

  1. #1
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    Exclamation Another relatively prime problem

    a) Show that if a, b, and c are integers with (a,b)=(a,c)=1, then (a,bc)=1.
    b) Use mathematical induction to show that if a1, a2,.....,an are integers, and b is another integer such that (a1,b)=(a2,b)=...=(an,b)=1, then(a1a2...an, b)=1


    Thanks for your help.
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  2. #2
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    Quote Originally Posted by suedenation View Post
    a) Show that if a, b, and c are integers with (a,b)=(a,c)=1, then (a,bc)=1.
    b) Use mathematical induction to show that if a1, a2,.....,an are integers, and b is another integer such that (a1,b)=(a2,b)=...=(an,b)=1, then(a1a2...an, b)=1


    Thanks for your help.
    a) (a,b) = 1 so the (unique) prime factorization of b contains NO primes in the prime factorization of a. Similarly (a,c) = 1 implies that the prime factorization of c contains no primes in the prime factorization of a. This means that the prime factorization of bc also contains no primes in the prime factorization of a. Thus (a,bc) = 1.

    b) You really want to "turn this around" and start with (a_1,b) = 1 and (a_2,b) = 1 implies (a_1*a_2,b) = 1. (After all, the symbol is symmetric.)

    Now, what about
    (a_1,b) = 1, (a_2,b) = 1, and (a_3,b) = 1.
    Let a_1*a_2 = d. Thus we need to prove
    (a_1*a_2*a_3,b) = (d*a_3,b) = 1.
    We know that (d,b) = (a_1*a_2,b) = 1 from a) and (a_3,b) = 1.
    Thus by a) we know that (d*a_3,b) = 1.

    Can you generalize this procedure?

    -Dan
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  3. #3
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    Quote Originally Posted by suedenation View Post
    a) Show that if a, b, and c are integers with (a,b)=(a,c)=1, then (a,bc)=1.
    Watch from die miester and learn.
    ---
    Let d=gcd(a,bc)
    thus,
    d|a and d|bc since, gcd(a,b)=d then,
    d|c (by Euclid's lemma )
    but,
    gcd(a,c)=1
    thus,
    the maximal nature of d is 1.
    Last edited by ThePerfectHacker; September 18th 2006 at 08:29 PM.
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