Let n be odd integer and m be even integer claim: \(\displaystyle{3}{m}^{{3}}+{2}{m}^{{2}}\) is odd.

We will prove this by contradiction.

On contrary suppose \(\displaystyle{3}{m}^{{3}}={2}{m}^{{2}}\) is not odd.

i.e. \(\displaystyle{3}{m}^{{3}}={2}{m}^{{2}}\) is even

\(\displaystyle\therefore{3}{m}^{{3}}\) is even

\(\displaystyle\therefore{m}^{{3}}\) is even

\(\displaystyle\therefore{m}\) is even

which is contradiction

\(\displaystyle\therefore\) our supposition is wrong

\(\displaystyle\therefore{3}{n}^{{3}}+{2}{m}^{{2}}\) is odd.

We will prove this by contradiction.

On contrary suppose \(\displaystyle{3}{m}^{{3}}={2}{m}^{{2}}\) is not odd.

i.e. \(\displaystyle{3}{m}^{{3}}={2}{m}^{{2}}\) is even

\(\displaystyle\therefore{3}{m}^{{3}}\) is even

\(\displaystyle\therefore{m}^{{3}}\) is even

\(\displaystyle\therefore{m}\) is even

which is contradiction

\(\displaystyle\therefore\) our supposition is wrong

\(\displaystyle\therefore{3}{n}^{{3}}+{2}{m}^{{2}}\) is odd.