đề thi toán cấp thành phố HD

C

chonhoi110

$A=x^5y - xy^5$
$=(x^5y-xy)-(xy^5-xy)$
$=yx(x^4-1)-xy(y^4-1)$
$=yx(x+1)(x-1)(x^2+1)-xy(y-1)(y+1)(y^2+1)$
Ta có: $yx(x+1)(x-1)(x^2+1)$ chia hết cho $6$($x(x+1)(x-1)$ là 3 số tự nhiên liên tiếp)
Tương tự: $xy(y-1)(y+1)(y^2+1)$chia hết cho $6$
\Rightarrow $A$ chia hết cho $6(1)$
$A=yx(x+1)(x-1)(x^2+1)-xy(y-1)(y+1)(y^2+1)$
$= yx( x- 1)(x + 1)(x^2-4+5)-xy(y-1)(y+1)(y^2-4+5)$
$= yx(x- 1)(x+ 1)[(x-2)(x+2)+5]-xy(y-1)(y+1)[(y-2)(y+2)+5]$
$= y(x- 1)(x+ 1)[x(x-2)(x+2)+5x]-x(y-1)(y+1)[y(y-2)(y+2)+5y]$
$= [xy(x - 1)(x + 1)(x-2)(x+2) +(x - 1)(x + 1)5x]-[xy(y-1)(y+1)(y-2)(y+2)+(y-1)(y+1)5y]$
$xy(x - 1)(x + 1)(x-2)(x+2) +(x - 1)(x + 1)5x$ chia hết cho 5
$xy(y-1)(y+1)(y-2)(y+2)+(y-1)(y+1)5y$ chia hết cho 5
\Rightarrow A chia hết cho $5 (2)$
Từ (1),(2) \Rightarrow $A$ chia hết cho $30$
 
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