\(n^4-10n^2+9=\left(n^4-9n^2\right)-\left(n^2-9\right)\)
\(=n^2.\left(n^2-9\right)-\left(n^2-9\right)=\left(n^2-1\right)\left(n^2-9\right)\)
\(=\left(n-1\right)\left(n+1\right)\left(n-3\right)\left(n+3\right)\)
Vì n lẻ \(\Rightarrow n=2k+1\)( \(k\inℤ\))
\(\Rightarrow n^4-10n^2+9=\left(2k+1-1\right)\left(2k+1+1\right)\left(2k+1-3\right)\left(2k+1+3\right)\)
\(=2k.\left(2k+2\right).\left(2k-2\right).\left(2k+4\right)\)
\(=16.k\left(k+1\right)\left(k-1\right)\left(k+2\right)\)
\(=16.\left(k-1\right).k.\left(k+1\right).\left(k+2\right)\)
Vì \(k-1\); \(k\); \(k+1\); \(k+2\)là 4 số nguyên liên tiếp
\(\Rightarrow\left(k-1\right).k.\left(k+1\right).\left(k+2\right)⋮24\)
\(\Rightarrow16.\left(k-1\right).k.\left(k+1\right).\left(k+2\right)⋮384\)
hay \(n^4-10n^2+9⋮384\)( đpcm )
