\(a^5-a=a\left(a^4-1\right)=a\left(a^2-1\right)\left(a^2+1\right).\)
\(=a\left(a-1\right)\left(a+1\right)\left(a^2-4+5\right)\)
\(=\left(a-1\right)a\left(a+1\right)\left(a^2-4\right)+5a\left(a-1\right)\left(a+1\right)\)
\(=\left(a-2\right)\left(a-1\right)a\left(a+1\right)\left(a+2\right)+5a\left(a-1\right)\left(a+1\right)\)
Vì \(\left(a-2\right)\left(a-1\right)a\left(a+1\right)\left(a+2\right)\)là tích của 5 số tự nhiên liên tiếp
\(\Rightarrow\left(a-2\right)\left(a-1\right)a\left(a+1\right)\left(a+2\right)\)\(⋮\)\(5\)
Mà \(5\)\(⋮\)\(5\)\(\Rightarrow5a\left(a-1\right)\left(a+1\right)\)\(⋮\)\(5\)
\(\Rightarrow\left(a-2\right)\left(a-1\right)a\left(a+1\right)\left(a+2\right)+5a\left(a-1\right)\left(a+1\right)\)\(⋮\)\(5\)
Hay \(a^5-a\)\(⋮\)\(5\)\(\left(đpcm\right)\)
