`A = (x+ 201y).(2x + 200y).(3x + 199y). ... .(10x + 192y)`
`A=(x-y+2.101y).(2x-2y+2.101y).(3x-3y+2.101y)...(10x-10y+2.101y)`
`A=(x-y+2.101y).[2(x-y)+2.101y]. [3(x-y)+2.101y]....[10(x-y)+2.101y]`
Vì `A\ \vdots\ 101`
`=>`$\left[\begin{array}{l}(x-y+2.101y)\ \vdots\ 101\\2(x-y)+2.101y\ \vdots\ 101\\ 3(x-y)+2.101y\ \vdots\ 101\\ ... \\10(x-y)+2.101y\ \vdots\ 101\end{array}\right.$
Giả sử `(x-y+2.101y)\ \vdots \ 101`
Vì `2.101y\ \vdots \ 101`
`=>(x-y)\ \vdots \ 101`
Ta có: `ƯCLN(2;101)=1; ƯCLN(3;101)=1;... ƯCLN(10;101)=1 `
`=>`$\begin{cases}2(x-y)\ \vdots \ 101\\3(x-y)\ \vdots \ 101\\... \\10(x-y)\ \vdots \ 101\end{cases}$
`=>A=(x-y+2.101y)..[2(x-y)+2.101y]. [3(x-y)+2.101y]....[10(x-y)+2.101y]\ \vdots\ \underbrace{101.\ 101\ . 101\ ...\ .101}_{10\ số\ 101}`
`=>A\ \vdots\ 101^{10}`
Vậy `A\ \vdots\ 101^{10}` (đpcm)