How to find the GCD(Greatest common Divisor) of the numbers given as algebraic expressions

Question

I am not able to establish that the GCD will not depend upon the value of $n$. Although by taking some initial values of $n$, the GCD is always 1, how do we prove that it is always 1 whatever be the value of $n$.

Answer 1

Let us denote the numbers by $A$ and $B$, where

$A=22n+1$

$B=33n+2$

Our first task is to establish a relation between $A$ and $B$ such that $n$ is eliminated.

$3A=66n+3$ and $2B=66n+4$

$\Rightarrow 2B-3A=1$

Now suppose GCD of $A,\,\,B$ is $k$, then $2B-3A$should be a multiple of $k$, but we have just proved that $2B-3A=1$, therefore $k=1$ only.