Question

cho ưcln(a,b)=1 . chứng minh rằng  ưcln(a+b;a)=1

Answer 1

Giả sử \(ƯCLN\left(a+b;a\right)\ne1\)

\(\RightarrowƯCLN\left(a+b;a\right)=d\left(d\inℕ^∗,d\ne1\right)\)

\(\Rightarrow\hept{\begin{cases}a+b⋮d\\a⋮d\end{cases}}\)

\(\Rightarrow b⋮d\)

\(\RightarrowƯCLN\left(a;b\right)=d\)

\(\Rightarrow\)Mâu thuẫn với \(ƯCLN\left(a;b\right)=d\)

\(\Rightarrow\)Điều giả sử là sai 

\(\RightarrowƯCLN\left(a+b;a\right)=1\)

\(\Rightarrowđpcm\)

Answer 2

\(ƯCLN\left(a+b,a\right)=ƯCLN\left(a,b\right)=1\)