\(a^5-a=a\left(a-1\right)\left(a+1\right)\left(a^2+1\right)=a\left(a-1\right)\left(a+1\right)\left(a^2-4+5\right)\)
\(\)\(=a\left(a-1\right)\left(a+1\right)\left(a^2-4\right)+5a\left(a-1\right)\left(a+1\right)\)
\(=a\left(a-1\right)\left(a+1\right)\left(a-2\right)\left(a+2\right)+5a\left(a-1\right)\left(a+1\right)\)
Chứng minh : \(a\left(a-1\right)\left(a+1\right)\left(a-2\right)\left(a+2\right)\)chia hết cho 5 và 6
\(a\left(a-1\right)\left(a+1\right)\)chia hết cho 6
Mà (5,6) = 1
\(\Rightarrow a\left(a-1\right)\left(a+1\right)\left(a-2\right)\left(a+2\right)\)chia hết cho 30
\(\Rightarrow5a\left(a-1\right)\left(a+1\right)\)chia hết cho 30
\(\Rightarrow a^5-a\) chia hết cho 30 (ĐPCM)
