If I am given to find a $\gcd$ and I can see a common divisor, can I divide it to make the problem smaller? if so what do I do with the common divisor?
In the case of $\gcd(2780,785)$ I can see the $5$ is a common divisor, and let $\gcd(556,157)=a$ what will be the $\gcd(2780,785)$?
Yes, you are right: if $k$ divides $a$ and $b$ then $\gcd(a,b)=k\gcd(a/k,b/k)$.
Another property of $\gcd$ which could be useful here is that $\gcd(a,b)=\gcd(a-kb,b)$ for $k\in \mathbb{Z}$. Hence $$\gcd(2780,785)=5\gcd(556,157)=5\gcd(556-3\cdot 157,157)=5\gcd(85,157).$$ Can you take it from here?
There are several strategies you can use.
1) You can use of course that $\gcd(ca,cb)=c\gcd(a,b)$ for everyy $a,b,c \in \mathbb N$.
2) Write $A = yB + C$ using the euclidean divion. Since you are looking for something dividing both $A$ and $B$ as you can see it must divide $C$. So you can say that $\gcd(A,B)=gcd(B,C)$.
Let's try to apply these two rules.
Applying $1)$ we get $\gcd(2780,556)=5\gcd(556,157)$.
I can't see another easy step so we can try to use euclidean division and write:
$$ 556 = 157\times 3 + 85 $$
thus using rule $2)$ we get $\gcd(556,157)=\gcd(157,85)$.
Let's try again, we write
$$ 85 \times 2 -157= 13 $$
thus $\gcd(157,85)=\gcd(85,13)=1$ because $13$ is a prime number which doesn't divide 85. Combining we get $$\gcd(2780,556)=5\gcd(556,157)=5\gcd(157,85)=5\gcd(85,13)=5$$
