Question

1) Tìm x, y biết : | x - 2013 | + | 1007 - \(\dfrac{1}{2}\)y | = 0

2) Cho n \(\in\) N , CMR : 9\(\times\)\(10^n+18⋮27\)

Answer 1

1. Với mọi x,y ta có :

\(\left\{{}\begin{matrix}\left|x-2013\right|\ge0\\\left|1007-\dfrac{1}{2}y\right|\ge0\end{matrix}\right.\)

Mà \(\left|x-2013\right|+\left|1007-\dfrac{1}{2}y\right|=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left|x-2013\right|=0\\\left|1007-\dfrac{1}{2}y\right|=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-2013=0\\1007-\dfrac{1}{2}y=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=2013\\y=2014\end{matrix}\right.\)

Vậy ...

2. Đặt :

\(H=9.10^n+18\)

Mà \(27=9.3\)

Ta có ;

\(A=9.10^n+18=9\left(10^n+2\right)\)

\(\Leftrightarrow A⋮9\)\(\left(1\right)\)

Lại có :

\(10^n+2=\left(10.......0\right)+2=100....02\)

\(\Leftrightarrow A⋮3\)\(\left(2\right)\)

Từ \(\left(1\right)+\left(2\right)\Leftrightarrow A⋮27\left(đpcm\right)\)