\(TH1:n\inℕ;n⋮2\)
\(\Rightarrow n+6⋮2\left(6⋮2\right)\)
\(\Rightarrow\left(n+3\right)\cdot\left(n+6\right)⋮2\)
\(TH2:n\inℕ;n⋮̸2\)
\(\Rightarrow n+3⋮2\)
\(\Rightarrow\left(n+3\right)\cdot\left(n+6\right)⋮2\)
\(TH1:n\inℕ;n⋮2\)
\(\Rightarrow n\left(n+1\right)⋮2\)
\(\Rightarrow n^2+n⋮2\)
\(\Rightarrow n^2+n+1⋮̸2\left(1⋮̸2\right)\)
\(TH2:n\inℕ;n⋮̸2\)
\(\Rightarrow n+1⋮2;n^2⋮̸2\)
\(\Rightarrow n^2+n+1⋮̸2\)