â) Gọi \(d=ƯCLN\left(4n-13;5n-16\right)\left(d\in N\right)\)
\(\Leftrightarrow\left\{{}\begin{matrix}4n-13⋮d\\5n-16⋮d\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}20n-65⋮d\\20n-64⋮d\end{matrix}\right.\)
\(\Leftrightarrow1⋮d\)
Vì \(d\in N;1⋮d\Leftrightarrow d=1\)
\(\LeftrightarrowƯCLN\left(4n-13;5n-16\right)=1\)
\(\Leftrightarrow\) Phân số \(\dfrac{4n-13}{5n-16}\) tối giản với mọi n
b) Gọi \(d=ƯCLN\left(5n-13;3n-8\right)\)
\(\Leftrightarrow\left\{{}\begin{matrix}5n-13⋮d\\3n-8⋮d\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}15n-39⋮d\\15n-40⋮d\end{matrix}\right.\)
\(\Leftrightarrow1⋮d\)
Vì \(d\in N;1⋮d\Leftrightarrow d=1\)
\(\LeftrightarrowƯCLN\left(5n-13;3n-8\right)=1\)
\(\Leftrightarrow\) Phân số \(\dfrac{5n-13}{3n-8}\) tối giản với mọi n
a) \(\dfrac{4n-13}{5n-16}\)
Đặt \(d=ƯCLN\left(4n-13;5n-16\right)\)
\(\Leftrightarrow\left\{{}\begin{matrix}4n-13⋮d\\5n-16⋮d\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}5\left(4n-13\right)⋮d\\4\left(5n-16\right)⋮d\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}20n-65⋮d\\20n-64⋮d\end{matrix}\right.\)
\(\Leftrightarrow\left(20n-65\right)-\left(20n-64\right)⋮d\)
\(\Leftrightarrow20n-65-20n+64⋮d\)
\(\Leftrightarrow-1⋮d\)
\(\Leftrightarrow d=\left\{1;-1\right\}\)
Vậy phân số \(\dfrac{4n-13}{5n-16}\) là phân số tối giản.
b) \(\dfrac{5n-13}{3n-8}\)
Đặt \(d=ƯCLN\left(5n-13;3n-8\right)\)
\(\Leftrightarrow\left\{{}\begin{matrix}5n-13⋮d\\3n-8⋮d\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}3\left(5n-13\right)⋮d\\5\left(3n-8\right)⋮d\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}15n-39⋮d\\15n-40⋮d\end{matrix}\right.\)
\(\Leftrightarrow\left(15n-39\right)-\left(15n-40\right)⋮d\)
\(\Leftrightarrow15n-39-15n+40⋮d\)
\(\Leftrightarrow1⋮d\)
\(\Leftrightarrow d=1\)
Vậy phân số \(\dfrac{5n-13}{3n-8}\) là phân số tối giản.
Gọi \(d\) là \(UCLN\left(4n-13;5n-16\right)\)
\(\Rightarrow\left\{{}\begin{matrix}4n-13⋮d\\5n-16⋮d\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}20n-65⋮d\\20n-64⋮d\end{matrix}\right.\)
\(\Rightarrow1⋮d\Rightarrow d=1\)
Vậy...
Gọi \(d\) là \(UCLN\left(5n-13;3n-8\right)\)
\(\Rightarrow\left\{{}\begin{matrix}5n-13⋮d\\3n-8⋮d\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}15n-39⋮d\\15n-40⋮d\end{matrix}\right.\)
\(\Rightarrow1⋮d\Rightarrow d=1\)
Vậy...
