Gọi \(d=ƯCLN\left(2n+3;4n+5\right)\)
\(\Leftrightarrow\left\{{}\begin{matrix}2n+3⋮d\\4n+5⋮d\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}4n+6⋮d\\4n+5⋮d\end{matrix}\right.\)
\(\Leftrightarrow1⋮d\)
\(\Leftrightarrow d=1\)
\(\LeftrightarrowƯCLN\left(2n+3;4n+5\right)=1\)
\(\Leftrightarrow2n+3;4n+5\) nguyên tố cùng nhau
\(\Leftrightarrow A=\dfrac{2n+3}{4n+5}\) tối giản với mọi n
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