Question

Chứng tỏ rằng với mọi số tự nhiên \(n\) thì tích \(\left(n+3\right)\left(n+6\right)\) chia hết cho 2 ?

Answer 1

Ta xét hai trường hợp

Nếu n chia hết cho 2 \(\Rightarrow n=2k\left(k\in n\right)\)

\(\Rightarrow\left(n+3\right)\left(n+6\right)=\left(2k+3\right)\left(2k+6\right)\)

\(=2k.2k+2k.6+3.2k+3.6\)

\(=2k^2+2k.6+2k.3+2.9\)

\(=2\left(k^2+6k+3k+9\right)⋮2\)

Nếu n chia cho 2 dư 1 \(\Rightarrow n=2k+1\)

\(\Rightarrow\left(2k+1+3\right)\left(2k+1+6\right)=\left(2k+4\right)\left(2k+7\right)\)

\(=2k.2k+2k.7+2k.4+4.7\)

\(=2k^2+2k.7+2k.4+2.14=2\left(k^2+7k+4k+14\right)⋮2\)

Vậy \(\left(n+3\right)\left(n+6\right)⋮2\left(n\in N\right)\)