How to find the g.c.d when I have another g.c.d

Question

$(a,b)=14$, ( ) as g.c.d

I want to find $$(4a+2b,6a+4b)=?$$

Is there some technique without doing it manually, I mean I though about it in such key:

$14|a$ adn $14|b$, we want to find such $k$ that: $k|4a+2b$ and $k|6a+4b$. $$$$ Then we want such $k$ that $k|2a+2b$ and $k|2a$(so k divides their sum wich is $4a+2b$), and $k|4a+4b$ and $k|2a$, in the same way, so I get 28 as the greatest.

But this is all only intuition and maybe not true. So how do I realy find it?

Answer 1

If $(a,b)=d,$

let $\dfrac aA=\dfrac bB=d\implies(A,B)=1$

Now $(4a+2b,6a+4b)=2d(2A+B,3A+2B)$

Let $d$ divides both $2A+B,3A+2B$

So, $k$ must divide

$(i)-3(2A+B)+2(3A+2B)=B$

and $(ii)2(2A+B)-(3A+2B)=A$

$\implies k$ must divide $(A,B)=1$