Question

[dangthihoa]

Cho Q(x) = ax²+bx+c (a;b;c ∈ ∈ Z ). Biết Q(x) chia hết ch0 2014 với mọi x ∈Z ∈ Z. Chứng minh rằng : 3a + 5b +7c chia hết cho 1007.

Answer 1

[Trần Tuấn Hoàng]

\(Q\left(0\right)=c⋮2014⋮1007\)

\(Q\left(1\right)=\left(a+b+c\right)⋮2014\Rightarrow\left(a+b\right)⋮2014\Rightarrow\left(2a+2b\right)⋮2014\)

\(Q\left(2\right)=\left(4a+2b+c\right)⋮2014\Rightarrow\left(4a+2b\right)⋮2014\)

\(\Rightarrow\left(4a+2b-2a-2b\right)⋮2014\)

\(\Rightarrow2a⋮2014\Rightarrow a⋮1007\Rightarrow b⋮1007\)

\(\Rightarrowđpcm\)