Ta có:
\(a^5-a\)
\(=a\left(a^4-1\right)\)
\(=a\left(a^2-1\right)\left(a^2+1\right)\)
\(=a\left(a-1\right)\left(a+1\right)\left[\left(a^2-4\right)+5\right]\)
\(=a\left(a-1\right)\left(a+1\right)\left(a^2-4\right)+5a\left(a-1\right)\left(a+1\right)\)
\(=a\left(a-1\right)\left(a+1\right)\left(a-2\right)\left(a+2\right)+5a\left(a-1\right)\left(a+1\right)\)
Vì \(a\left(a-1\right)\left(a+1\right)\left(a-2\right)\left(a+2\right)⋮30\)
\(5a\left(a-1\right)\left(a+1\right)⋮30\)
\(\Rightarrow a^5-a⋮30\)
`=a(a4−1)`
`=a(a2−1)(a2+1)`
`=a(a−1)(a+1)[(a2−4)+5]`
`=a(a−1)(a+1)(a2−4)+5a(a−1)(a+1)`
`=a(a−1)(a+1)(a−2)(a+2)+5a(a−1)(a+1)`
Vì `a(a−1)(a+1)(a−2)(a+2)⋮30`
`5a(a−1)(a+1)⋮30`
`⇒a5−a⋮30`
