Question

Tìm tất cả các số nguyên để phân số \(\frac{18n+3}{21n+7}\)là phân số tối giản.

Answer 1

Gọi \(d\inƯC\left(18n+3;21n+7\right)\)

\(\Rightarrow\left\{\begin{matrix}18n+3⋮d\\7\left(18n+3\right)⋮d\end{matrix}\right.\)

\(\Rightarrow\left\{\begin{matrix}21n+7⋮d\\6\left(21n+7\right)⋮d\end{matrix}\right.\)

\(\Rightarrow6\left(21n+7\right)-7\left(18n+3\right)⋮d\)

\(\Rightarrow21⋮d\Rightarrow d=\left\{3;7\right\}\)

Nếu \(k=3\Rightarrow21n+7\)\(⋮̸\)\(3\) (loại)

Nếu \(k=7\Rightarrow21n+7⋮7\Rightarrow21n-3n+3⋮7\)

\(\Rightarrow3-3n⋮7\Rightarrow3-3n=7k\left(k\in N\right)\)

\(\Rightarrow1-n=\frac{7k}{3}\Rightarrow n=1-\frac{7k}{3}\Rightarrow k=0;n=1\)

Vậy \(n=1\)