Question

+Chứng minh:
\(n^4-10n^2+9\text{ }⋮\text{ }384\text{ }v\text{ới }n\text{ }l\text{ẻ }\left(n\in Z\right) \)

Answer 1

Vì n lẻ nên n=2k+1

\(n^4-10n^2+9\)

\(=\left(n^2-1\right)\left(n^2-9\right)\)

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

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

\(=2k\cdot\left(2k+2\right)\cdot\left(2k-2\right)\cdot\left(2k+4\right)\)

\(=16k\left(k+1\right)\left(k-1\right)\left(k+2\right)\)

Vì k-1;k+1;k;k+2 là bốn số liên tiếp

nên \(\left(k-1\right)\cdot k\cdot\left(k+1\right)\cdot\left(k+2\right)⋮4!=24\)

\(\Leftrightarrow16k\left(k+1\right)\left(k-1\right)\left(k+2\right)⋮384\)