Question

cho a,b,c thuộc Z thoã mãn a+b+c=0

cmr a^5+b^5+c^5 chia hết cho 5abc

Answer 1

Ta có :

\(a+b+c=0\)

\(\Leftrightarrow a+b=-c\)

\(\Leftrightarrow\left(a+b\right)^5=-c^5\)

\(\Leftrightarrow a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5=-c^5\)

\(\Leftrightarrow a^5+b^5+c^5=-5ab\left(a^3+b^3+2a^2b+2ab^2\right)\)

\(\Leftrightarrow a^5+b^5+c^5=-5ab\left[\left(a^3+b^3\right)+2ab\left(a+b\right)\right]\)

\(\Leftrightarrow a^5+b^5+c^5=-5ab\left(a+b\right)\left(a^2+ab+b^2\right)\)

\(\Leftrightarrow a^5+b^5+c^5=-5abc\left(a^2+ab+b^2\right)\)

\(\Leftrightarrow a^5+b^5+c^5\) chia hết cho \(5abc\left(đpcm\right)\)