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Math Help - coprime problem

  1. #1
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    warangal
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    coprime problem

    pls solve this problem for me as m not also to solve it...

    question: suppose a and b are relatively prime. prove that ab and a+b are relatively prime...
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  2. #2
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    Jaipur,Rajasthan
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    Re: coprime problem

    suppose a=3
    b=5
    ab=5*3=15
    a+b=8

    so a+b is not a relatively prime.
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  3. #3
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    Re: coprime problem

    Quote Originally Posted by kraj8995 View Post
    so a+b is not a relatively prime.
    You cannot say "relatively prime" about one number.
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  4. #4
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    Re: coprime problem

    two numbers a,b are said to be coprime if gcd(a,b)=1
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  5. #5
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    Re: coprime problem

    one could suppose that a counter-example existed. if so then since d > 1 divides ab and a+b, there is some prime p that divides d. but p, being prime, must divide a or b.

    suppose it were a, that is a = kp. since p also divides a+b (since d does), a+b = rp. hence b = a+b - a = rp - kp = (r-k)p, so p divides b.

    or, suppose it were b, so that b = mp. then a = a+b - b = rp - mp = (r-m)p, and so a is likewise divisible by p.

    either way, assuming that gcd(ab,a+b) > 1 leads to a prime p dividing a and b. but gcd(a,b) = 1, so how can this be possible?
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