Question

[x+y][x+2y][x+3y]x+4y]+\(^{y^4}\)la so chinh phuong

Cho \(a_1\),\(a_2\),......,\(a_{2016}\)la cac so tu nhien co tong chia het cho 3

CM: \(a_1^3+a^3_2+....+a^3_{16}\)Chia het cho 3

Answer 1

a) Đặt \(A=\left(x+y\right)\left(x+2y\right)\left(x+3y\right)\left(x+4y\right)+y^4\)

\(\Rightarrow A=\left(x+y\right)\left(x+4y\right)\left(x+2y\right)\left(x+3y\right)+y^4\)

\(=\left(x^2+5xy+4y^2\right)\left(x^2+5xy+6y^2\right)+y^4\)

Đặt \(x^2+5xy+5y^2=t\)

\(\Rightarrow A=\left(t+y^2\right)\left(t-y^2\right)+y^4=t^2-y^4+y^4\)

         \(=t^2=\left(x^2+5xy+5y^2\right)^2\)là số chính phương ( đpcm )