Question

CMR: phân số tối giản :

\(\dfrac{2n+1}{2n\left(n+1\right)}\)

Answer 1

\(\dfrac{2n+1}{2n\left(n+1\right)}=\dfrac{2n+1}{2n^2+2n}\)

Gọi \(d=ƯCLN\left(2n+1;2n^2+2n\right)\left(d\in N\right)\)

\(\Leftrightarrow\left\{{}\begin{matrix}2n+1⋮d\\2n^2+2n⋮d\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2n^2+n⋮d\\2n+1⋮d\end{matrix}\right.\)

\(\Leftrightarrow n⋮d\)

Mà \(2n+1⋮d\)

\(\Leftrightarrow\left\{{}\begin{matrix}2n⋮d\\2n+1⋮d\end{matrix}\right.\)

\(\Leftrightarrow1⋮d\)

\(\Leftrightarrow d=1\)

\(\LeftrightarrowƯCLN\left(2n+1;2n\left(n+1\right)\right)=1\)

\(\Leftrightarrow\) Phân số \(\dfrac{2n+1}{2n\left(n+1\right)}\) là phân số tối giản