Question

Chứng minh phân số:

\(\frac{4n+3}{5n+4}\) tối giản với mọi \(n\in\) N*

Answer 1

Gọi ƯCLN(4n+3;5n+4)=d (d\(\in\)Z; d\(\ne\)0)

\(\Rightarrow\) \(\left(4n+3\right)⋮d\) \(và\) \(\left(5n+4\right)⋮d\)

\(\Rightarrow5\left(4n+3\right)⋮d\) \(và\) \(4\left(5n+4\right)⋮d\)

\(\Rightarrow\left(20n+15\right)⋮d\) \(và\) \(\left(20n+16\right)⋮d\)

\(\Rightarrow\left(20n+16\right)-\left(20n+15\right)\)\(⋮d\)

\(\Rightarrow1⋮d\) \(\Rightarrow d\inƯ\left(1\right)\)

mà Ư(1)={1;-1}

\(\Rightarrow\) \(d\in\left\{1;-1\right\}\)

\(Khi\) \(đó\) \(phân\) \(số\) \(\frac{4n+3}{5n+4}\) \(là\) \(phân\) \(số\) \(tối\) \(giản\)

Vậy ...........