# Using "Proof by Contraposition",sho that: if n is any odd integer and m is any even integer, then, 3m^3+2m^2 is odd.

Question
Upper Level Math
Using "Proof by Contraposition",sho that: if n is any odd integer and m is any even integer, then, $$\displaystyle{3}{m}^{{3}}+{2}{m}^{{2}}$$ is odd.

2021-02-06
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
$$\displaystyle\therefore$$ our supposition is wrong
$$\displaystyle\therefore{3}{n}^{{3}}+{2}{m}^{{2}}$$ is odd.

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