I'm not sure if it goes here or the section beyond calculus, so I'm just putting it here because it doesn't involve any calculus.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Suppose that (a,b)=1 [Greatest Common Divisor=1] and (a,c)=1. Does (bc, a)=1?

2. Relevant equations

(a,b)=d=au+bv, where u and v are integers and d is the greatest common divisor of a and b.

3. The attempt at a solution

OK, so I'm taking both of the facts I already know, that (a,b)=1 and (a,c)=1 and turning them into a useful equation:

1=au+bv, and 1=am+cn where u,v,m, and n are all integers. I know that my end goal is to find a(q)+bc(r)=1 However, I can't seem to find a way to get there. The closest I've gotten is by setting the two equal:

au+bv=am+cn, and them multiplying by bc to get:

abcu-abcm=bc^{2}n+b^{2}cv

Then I factor and move them to the same side to find:

0=a(bcm-bcu)+bc(cn+bv)

Which is not what I need.

Am I going about this the completely wrong way? I have a feeling I'm mistaking a basic part of the proof, as a similar problem is also giving me trouble.

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# Homework Help: Proofs Involving Greatest Common Divisors

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