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.
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!