a/ Để phân số \(\dfrac{n+1}{n-2}\) có giá trị nguyên thì :
\(n+1⋮n-2\)
Mà \(n-2⋮n-2\)
\(\Leftrightarrow3⋮n-2\)
\(\Leftrightarrow n-2\inƯ\left(3\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}n-2=1\\n-2=3\\n-2=-1\\n-2=-3\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}n=3\\n=5\\n=1\\n=-1\end{matrix}\right.\)
Vậy ..
b/ Gọi \(d=ƯCLN\left(12n+1;30n+2\right)\)
\(\Leftrightarrow\left\{{}\begin{matrix}12n+1⋮d\\30n+2⋮d\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}60n+5⋮d\\60n+4⋮d\end{matrix}\right.\)
\(\Leftrightarrow1⋮d\)
\(\Leftrightarrow d=1\)
\(\LeftrightarrowƯCLN\left(12n+1;30n+2\right)=1\)
\(\Leftrightarrow\) Phân số \(\dfrac{12n+1}{30n+2}\) tối giản với mọi n
