a) Gọi ƯCLN( 2n+3; 4n+8)=d \(\left(d\in N\cdot\right)\)
\(\Rightarrow\left\{{}\begin{matrix}\left(2n+3\right)⋮d\\\left(4n+8\right)⋮d\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}\left(4n+6\right)⋮d\\\left(4n+8\right)⋮d\end{matrix}\right.\)
\(\Rightarrow2⋮d\Leftrightarrow d\inƯ\left(2\right)=\left\{1;2\right\}\)
Nếu \(d=2\) thì \(\left(2n+3\right)⋮2\), vô lý
\(\Rightarrow d=1\RightarrowƯCLN\left(2n+3;4n+8\right)=1\)
Vậy ta có đpcm.
b) Gọi ƯCLN(7n+3;5n+2)=d \(\left(d\in N\cdot\right)\)
\(\Rightarrow\left\{{}\begin{matrix}\left(7n+3\right)⋮d\\\left(5n+2\right)⋮d\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}5\left(7n+3\right)⋮d\\7\left(5n+2\right)⋮d\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\left(35n+15\right)⋮d\\\left(35n+14\right)⋮d\end{matrix}\right.\)
\(\Rightarrow1⋮d\Leftrightarrow d=1\)
nên ƯCLN(7n+3;5n+2)=1
Vậy ta có đpcm.
