Question

[Nguyễn Đoàn Hồng Thái]

CMR nếu x;y∈N thì:

3x+7y⋮31 khi và chỉ khi 16x+27y⋮31

Answer 1

[ngonhuminh]

A)\(\left(3x+7y\right)⋮31\Rightarrow\left(15x+35y\right)⋮31\Rightarrow\left[31\left(x+y\right)+31y-\left(16x+27y\right)\right]⋮31\)

\(x;y\in N\Rightarrow\left\{{}\begin{matrix}31\left(x+y\right)⋮31\\31y⋮31\\\end{matrix}\right.\) \(\Rightarrow\left(16x+27y\right)⋮31\)

B)

\(\left(16x+27y\right)⋮31\Rightarrow\left[31x+62y-\left(15x+35y\right)\right]⋮31\Rightarrow\left[31x+62y-5\left(3x+7y\right)\right]⋮31\)\(x;y\in N\Rightarrow\left\{{}\begin{matrix}31x⋮31\\62y⋮31\\5⋮̸31\end{matrix}\right.;\left(5;31\right)=1}\)\(x;y\in N\Rightarrow\left\{{}\begin{matrix}31\left(x+y\right)⋮31\\31y⋮31\\5⋮̸31\end{matrix}\right.\) ; \(\left(5;31\right)=1\Rightarrow\left(3x+7y\right)⋮31\)

từ (A) và (B) : \(\left(3x+7y\right)⋮31\Leftrightarrow\left(16x+27y\right)⋮31\) =>dpcm \(\Leftrightarrow dccm\)