Gọi d ∈ ƯCLN(5n + 6, 3n +1)
Để phân số \(\frac{5n+6}{3n+1}\) rút gọn được thì \(\left\{{}\begin{matrix}5n+6⋮d\\3n+1⋮d\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}3\left(5n+6\right)⋮d\\5\left(3n+1\right)⋮d\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}15n+18⋮d\\15n+5⋮d\end{matrix}\right.\)
\(\Rightarrow15n+18-\left(15n+5\right)⋮d\)
\(\Rightarrow15n+18-15n-5⋮d\)
\(\Rightarrow13⋮d\)
\(\Rightarrow d\inƯ\left(13\right)=\left\{1;13\right\}\)
Để phân số \(\frac{5n+6}{3n+1}\) rút gọn được thì d = 13
\(\Rightarrow3n+1⋮13\)
\(\Rightarrow3n+1+12-12⋮13\)
\(\Rightarrow3n-12+13⋮13\)
\(\Rightarrow3n-12⋮13\)
\(\Rightarrow3\left(n-4\right)⋮13\)
\(\Rightarrow\left(n-4\right)⋮13\) vì (3,13) = 1
\(\Rightarrow n-4=13k\)
\(\Rightarrow n=13k+4\)
ta có: \(60< n< 100\)
\(\Rightarrow60< 13k+4< 100\)
\(\Rightarrow56< 13k< 96\)
\(\Rightarrow5\le k\le7\)
\(\Rightarrow k\in\left\{5;6;7\right\}\)
\(\Rightarrow n\in\left\{69;82;95\right\}\)
