Question

Chứng tỏ rằng:2n+1 và 3n+1 là 2 số nguyên tố cùng nhau

Answer 1

Gọi \(d=ƯC\left(2n+1;3n+1\right)\)

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

\(\Rightarrow3\left(2n+1\right)-2\left(3n+1\right)⋮d\)

\(\Rightarrow6n+3-6n-2⋮d\)

\(\Rightarrow1⋮d\)

\(\Rightarrow d=1\Rightarrow2n+1\) và \(3n+1\) nguyên tố cùng nhau