Question

[lyhaiquan]

chung minh rang voimoi so nguyen n thi a =n^3(n^2-7)^2-36n chia het cho 105

Answer 1

[Pham Van Hung]

\(A=n^3\left(n^2-7\right)^2-36n\)

\(=n\left[n^2\left(n-7\right)^2-36\right]\) 

\(=n\left[\left(n^3-7n\right)^2-6^2\right]\)

\(=n\left(n^3-7n-6\right)\left(n^3-7n+6\right)\)

\(=n\left[n^3+n^2-n^2-n-6n-6\right].\left[n^3-n^2+n^2-n-6n+6\right]\)

\(=n\left[n^2\left(n+1\right)-n\left(n+1\right)-6\left(n+1\right)\right]\left[n^2\left(n-1\right)+n\left(n-1\right)-6\left(n-1\right)\right]\)

\(=n\left(n+1\right)\left(n^2-n-6\right)\left(n-1\right)\left(n^2+n-6\right)\)

\(=n\left(n+1\right)\left[n\left(n-3\right)+2\left(n-3\right)\right]\left(n-1\right)\left[n\left(n+3\right)-2\left(n+3\right)\right]\)

\(=n\left(n+1\right)\left(n-3\right)\left(n+2\right)\left(n-1\right)\left(n+3\right)\left(n-2\right)\)

\(=\left(n-3\right)\left(n-2\right)\left(n-1\right)n\left(n+1\right)\left(n+2\right)\left(n+3\right)\)

Vì A là tích của 7 số nguyên liên tiếp nên A chia hết cho 3,5 và 7

\(\Rightarrow A⋮\left(3.5.7\right)\Rightarrow A⋮105\)

Chúc bạn học tốt.

Answer 2

[lyhaiquan]

n^2(n-7)=(n^3-7n^2)^2 chu ban