TITLE: Prove that for any integer $n$, the fraction $\frac{3n+2}{4n+3}$ is reduced.
QUESTION [10 upvotes]: I need help with this question.
Prove that for any integer $n$, the fraction $\frac{3n+2}{4n+3}$ is reduced.

REPLY [16 votes]: Note that 
$$3(4n+3)-4(3n+2)=1.$$
So any integer that divides both $4n+3$ and $3n+2$ must divide $1$. And nothing $\gt 1$ divides $1$, so the greatest common divisor of $4n+3$ and $3n+2$ is $1$.