Addition or subtraction in GCD and LCM

Question

Suppose that we have two integers $a$ and $b$. Now say that $G = \gcd(a,b)$ and $L = \mathrm{lcm}(a,b)$. Now the value of $G$ and $L$ is given and another integer $c$'s value is given. How can we find $\gcd(a+c,b+c)$ and $\mathrm{lcm}(a+c,b+c)$ from $G$, $L$ and $c$?

What if we have $n$ arbitrary numbers. I know the GCD and LCM of those numbers but not the actual values of those numbers. Now I want to add $c$ with all of those numbers, what will be the new GCD, LCM of those numbers?

Answer 1

Of course $\gcd (ka,kb) = |k|\gcd (a,b)$ for some integer $k \ne 0$ is easy to prove, but I do not think is a generalization you seek because this would mean the the gcd of any two numbers is somehow generated by $\gcd (a,b)$, where $a$ and $b$ are arbitrary. That does not seem likely. Unless $a$ and $b$ are coprime, which is trivial. The same goes for lcm.