[Toán 8]Chứng minh chia hết

K

kool_boy_98

$x^5y-xy^5$
$=xy(x^4-1-y^4+1)$
$=xy(x^4-1)-xy(y^4-1)$
$=xy(x-1)(x+1)(x^2+1)-xy(y-1)(y+1)(y^2+1)$

Ta thấy: $xy(x-1)(x+1)(x^2+1)$ chia hết cho $2;3;5$ hay $xy(x-1)(x+1)(x^2+1) \vdots 30$
Tương tự: $xy(y-1)(y+1)(y^2+1) \vdots 30$
$\Longrightarrow [xy(x-1)(x+1)(x^2+1)-xy(y-1)(y+1)(y^2+1)] \vdots 30$
Hay: $(x^5y-xy^5) \vdots 30$.

^^~.
 
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