What would the GCD of $3n + 2$ and $4n + 3$ be using Euclid's algorithm
Since the remainder is $\pm 1$, the GCD is $1$.
Alternatively, note that the greatest common divisor must divide both numbers, as well as their difference. So if $d$ is the GCD, $d|(n+1)$. But $(n+1)$ cannot share any factors with $(4n+3)=4\cdot(n+1)-1$.