Let d be the common divisor of 12n + 1 and 30n + 2.We have :

\(12n+1⋮d\Rightarrow5\left(12n+1\right)⋮d\Rightarrow60n+5⋮d\)

\(30n+2⋮d\Rightarrow2\left(30n+2\right)⋮d\Rightarrow60n+4⋮d\)

\(\Rightarrow\left(60n+5\right)-\left(60n+4\right)⋮d\Rightarrow1⋮d\Rightarrow d=\pm1\)

So \(\dfrac{12n+1}{30n+2}\) is a fraction in lowest terms with any \(n\in Z\)