Quote:

2) If $\displaystyle a|(b+c)$ and $\displaystyle (b,c)=1$, prove that $\displaystyle (a,b) = 1 = (a,c)$

Suppose $\displaystyle d|a$ and $\displaystyle d|b$

$\displaystyle a=du$ for some u in Z

$\displaystyle b=dv$ for some v in Z

$\displaystyle a|(b+c)$

$\displaystyle b+c=aw$

$\displaystyle c=aw-b$

$\displaystyle c=duw-dv$

$\displaystyle c=d(uw-v)$

$\displaystyle d|c$

Now, how do I say that (a,b) = 1 = (a,c) ?

Thanks for any help.

we will use the same plan of attack on this guy. we need to verify the two conditions for 1 in regards to both $\displaystyle (a,b)$ and $\displaystyle (a,c)$.