<s>[INST] Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$. Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$.
I know there exists some slick technique to simplify this problem. Any hints are greatly appreciated. 
 [/INST] HINT 
let $d=(2a^2+29a+65,a+13)$. We notice that $-13$ is a root of $2a^2+29a+39$. Another root is $-\frac32$
So we have $2a^2+29a+65=(a+13)(2a+3)+26$
Thus $d=((a+13)(2a+3)+26,a+13)\Rightarrow$ $d|(a+13)(2a+3)+26, d|a+13\Rightarrow d|(a+13)(2a+3)\Rightarrow d|26$
</s>