a) Đặt \(A=\frac{8n+193}{4n+3}=\frac{2\left(4n+3\right)+187}{4n+3}=2+\frac{187}{4n+3}\)
Để \(2+\frac{187}{4n+3}\) có giá trị tự nhiên thì
\(187⋮4n+3\)
\(\Rightarrow4n+3\inƯ\left(187\right)=\left\{17;11;187\right\}\)
\(4n+3=11\Rightarrow n=2\)
\(4n+3=187\Rightarrow n=46\)
\(4n+3=17\Rightarrow n=\frac{14}{4}\) (loại)
\(\Rightarrow n=\left\{2;46\right\}\)
b) Gọi \(ƯCLN\left(8n+193;4n+3\right)=d\)
\(\Rightarrow\left(8n+193;4n+3\right)⋮d\)
\(\Rightarrow\left(8n+193\right)-2\left(4n+3\right)\)
\(\Rightarrow\left(8n+193\right)-\left(8n+6\right)⋮d\)
\(\Rightarrow187⋮d\) mà \(A\) là phân số tối giản \(\Rightarrow A\ne d\)
\(\Rightarrow n\ne11k+2\left(k\in N\right)\)
\(\Rightarrow n\ne17m+12\left(m\in N\right)\)
