Ta có :
\(m⋮2\Leftrightarrow m=2k\left(k\in N\right)\)
\(\Leftrightarrow m^3+20m=\left(2k\right)^3+20.2k\)
\(=8k^3+40k\)
\(=8k\left(k^2+5\right)\)
Cần chứng minh \(k\left(k^2+5\right)⋮6\)là xong.
+ nếu \(k\) chẵn \(\Leftrightarrow k\left(k^2+5\right)⋮2\)
+ nếu \(k\) lẻ\(\Leftrightarrow k^2\) lẻ \(\Leftrightarrow k^2+5\) chẵn \(\Leftrightarrow k\left(k^2+5\right)⋮2\)
Vậy \(k\left(k^2+5\right)⋮2\)
+ nếu \(k⋮3\) \(\Leftrightarrow k\left(k^2+5\right)⋮3\)
+ nếu \(k=3k_1+1\)\(\Leftrightarrow k^2+5=\left(3k_1+1\right)^2+5=9k_1+6k+6⋮3\)
+ nếu \(k=3k_2+2\) \(\Leftrightarrow k^2+5=\left(3k_2+2\right)^2+5=9k^2_2+12k_2+9⋮3\)
Vậy \(k\left(k^2+5\right)⋮3\)
=>dpcm
Đúng 0
Bình luận (1)
