Gọi \(d=ƯCLN\left(n^4+3n^2+1;n^3+2n\right)\)

\(\Leftrightarrow\left\{{}\begin{matrix}n^4+3n^2+1⋮d\^3+2n⋮d\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}n^4+3n^2+1⋮d\^4+2n^2⋮d\end{matrix}\right.\)

\(\Leftrightarrow n^2+1⋮d\)

Mà \(n^3+2n⋮d\)

\(\Leftrightarrow\left\{{}\begin{matrix}n^3+n⋮d\^3+2n⋮d\end{matrix}\right.\)

\(\Leftrightarrow n⋮d\)

Mà \(n^2+1⋮d\)

\(\Leftrightarrow1⋮d\)

\(\Leftrightarrow d=1\)

\(\LeftrightarrowƯCLN\left(n^4+3n^2+1;n^3+2n\right)=1\)

\(\Leftrightarrow n^4+3n^2+1;n^3+2n\) nguyên tố cùng nhau

CMR các cặp số sau đây là số nghuen tố cùng nhau :

9n+4 và 12n+5

Gọi \(d=ƯCLN\left(9n+4;12n+5\right)\)

\(\Leftrightarrow\left\{{}\begin{matrix}9n+4⋮d\\12n+5⋮d\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}36n+16⋮d\\36n+15⋮d\end{matrix}\right.\)

\(\Leftrightarrow1⋮d\)

\(\Leftrightarrow d=1\)

\(\LeftrightarrowƯCLN\left(9n+4;12n+5\right)=1\)

\(\Leftrightarrow9n+4;12n+5\) nguyên tố cùng nhau