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Math Help - GCD true or false?

  1. #1
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    GCD true or false?

    Assume gcd(a,b)=1 and gcd (a,c)=1.

    True or false? If true, give a proof. If false, give a counterexample.

    a) gcd (bc,a)=1

    b) gcd (ab,c)=1

    I am pretty sure b) is false. We could do (a,b)=(3,5)=1 and (a,c)=(3,5)=1, but (ab,c)=(15,5)=5...

    I am not sure about a) though. I think it is true, but do not know how to prove it.
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  2. #2
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    Quote Originally Posted by twittytwitter View Post
    Assume gcd(a,b)=1 and gcd (a,c)=1.
    True or false? If true, give a proof. If false, give a counterexample.
    a) gcd (bc,a)=1
    I think it is true, but do not know how to prove it.
    Here is a way to proceed.
    a & b have no common prime factors.
    a & c have no common prime factors.
    The prime factors of bc must be prime factors of b or c.
    Can you finish?
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  3. #3
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    Let me try

    gcd(a,c)=1 and gcd(a,b)=1 and let gcd(a, bc)=x.
    Choose any prime factor p of x. So x divides a and b*c, so p also must divide these. Since p is prime, it divides b or it divides c. Let it divide b. But since gcd(a,b)=1, we must have p=1, and the same argument applies if p divides c.
    So no prime factors of x exist, and so x must be 1, and so gcd(a,bc)=1

    Does that look correct? And is the other part okay?

    Thanks!
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  4. #4
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    Quote Originally Posted by twittytwitter View Post
    gcd(a,c)=1 and gcd(a,b)=1 and let gcd(a, bc)=x.
    Choose any prime factor p of x. So x divides a and b*c, so p also must divide these. Since p is prime, it divides b or it divides c. Let it divide b. But since gcd(a,b)=1, we must have p=1, and the same argument applies if p divides c.
    So no prime factors of x exist, and so x must be 1, and so gcd(a,bc)=1
    It maybe a bit rough. But it certainly works.
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