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.

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
asked 2021-02-05
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.

Answers (1)

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
which is contradiction
\(\displaystyle\therefore\) our supposition is wrong
\(\displaystyle\therefore{3}{n}^{{3}}+{2}{m}^{{2}}\) is odd.
0

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