$\text{lcm} (a, b)=\text{lcm} (a+c, b+c)$ 
Can $\text{lcm} (a, b)=\text{lcm} (a+c, b+c)$ for natural $a, b, c$?

I've tried writing out all divisors of $a, b, c$ like $p_1 p_2$ etc. And tried that maybe if $a+c> a$ and $b+c> b$ the $\text{lcm}$ must be greater. 

This is not true in general as we see
lcm$(3,4)=12$ whereas lcm$(3+3,4+3)=42$
One more note, as you stated that $a,b,c$ are all naturals hence $c>0$ and thus always $a+c>a,b+c>b$ But you see that lcm$(7,9)=63$ whereas lcm$(7+1,9+1)=40<63$
