Question

cho a, b thuoc N : a² + b² chia het a x b . Tinh P= a² + b² trên ab

 

Answer 1

Gọi d là \(ƯCLN\left(a,b\right)\)

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

\(a=dm,b=dn\)với \(m,n\inℕ,\left(m,n\right)=1\)

\(\Rightarrow\hept{\begin{cases}a^2+b^2=d^2\left(m^2+n^2\right)\\ab=d^2mn\end{cases}}\)

\(a^2+b^2⋮ab\)

\(\Rightarrow d^2\left(m^2+n^2\right)⋮d^2mn\)

\(\Rightarrow m^2+n^2⋮mn\)

Do \(\left(m,n\right)=1\)

\(\Rightarrow\hept{\begin{cases}m^2+n^2⋮m\\m^2+n^2⋮n\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}n⋮m\\m⋮n\end{cases}}\)

\(\Rightarrow m=n\)

Mà \(\left(m,n\right)=1\)

\(\Rightarrow m=n=1\)

\(\Rightarrow P=\frac{a^2+b^2}{ab}=\frac{1+1}{1}=\frac{2}{1}=2\)

Vậy P=2