Proof by contrapositive
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Use a contrapositive proof to prove that if \(n\) be an integer and \(n\) is even then \(n^3+5\) is odd.
$$\begin{aligned}
&\text{The contrapositive statement is :}\\
&\text{If}\ n\ \text{is odd then}\ n^{3}+5\ \text{is even.}\\
&\begin{aligned}
\text{Let}\ n=2 k+1\ \text{then}\ n^{3}+5&=(2 k+1)^{3}+5\\
&=8 k^{3}+12 k^{2}+6 k+1+5\\
&=2(4 k^{3}+6 k^{2}+3 k+3)\end{aligned}\\
&\therefore\ n^{3}+5\ \text{is even}
\end{aligned}$$
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