Question

Chứng minh 

n (n² + 1) (n²+4) chia hết cho 5

Answer 1

\(A=n\left(n^2+1\right)\left(n^2+4\right)\)

+ n  ≡ 0 (mod 5)
=> n⋮5 => A⋮5

+ n  ≡ 1 (mod 5)
=> n²  ≡ 1 (mod 5)
=> n² + 4⋮5
=> A⋮5

+ n ≡ 2 (mod 5)
 => n² ≡ 4 (mod 5)
=> n² + 1⋮5
=> A⋮5

+ n ≡ 3 (mod 5)
 => n² ≡ 3²  ≡ 4 (mod 5)
=>n² + 1⋮5
=> A⋮5

+ n ≡ 4 (mod 5)
=>n² ≡ 4² ≡ 1 (mod 5)
=> n² +4⋮5
=> A⋮5

Vậy A luôn chia hết cho 5.