Question

cm a5 -a chia hết cho 30 với a thuộc Z

Answer 1

\(a^5-a=a\left(a-1\right)\left(a+1\right)\left(a^2+1\right)=a\left(a-1\right)\left(a+1\right)\left(a^2-4+5\right)\)

\(\)\(=a\left(a-1\right)\left(a+1\right)\left(a^2-4\right)+5a\left(a-1\right)\left(a+1\right)\)

\(=a\left(a-1\right)\left(a+1\right)\left(a-2\right)\left(a+2\right)+5a\left(a-1\right)\left(a+1\right)\)

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)