Question

If $a|\left(b+c\right)$and $\left(b,c\right)=1$, prove that $\left(a,b\right)=1=\left(a,c\right)$.

Short answer

It is proved that $\left(a,b\right)=1=\left(a,c\right)$.

Step-by-step solution

Use given part

It is given that $a|\left(b+c\right)$ and $\left(b,c\right)=1$.

Let $d=\left(a,b\right)$.

For $a|\left(b+c\right)$, let there exist an integer $w$. Then $b+c=aw$.

Prove a,b=1=a,c

From step 1, $c$ can be written as a linear combination of $a$ and $b$, so $d|c$.

Then, $d|\left(b,c\right)$ will imply that $d=1$.

Use d=1 in $d=\left(a,b\right)$.

$1=\left(a,b\right)$

Similarly, we will get $1=\left(a,c\right)$.

Hence, it is proved that $\left(a,b\right)=1=\left(a,c\right)$.