Proof by Contradiction - Questions
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Use proof by contradiction to prove that if \(3n+5\) is odd then \(n\) is even.Â
$$\begin{aligned}
&\text{Assume for a contradiction that is}\ 3 n+5\ \text{is odd}\\
&\text{then}\ n\ \text{is odd.}\\
&\text{Since}\ n\ \text{is odd then}\ n=2k+1\ \text{for some integer k.}\\
&\begin{aligned} 3 n+5 &=3(2 k+1)+5 \\
&=6 k+3+5 \\
&=6 k+8 \\
&=2(3 k+4) \\
&=2 m \quad m=3 k+4 \text { an integer } \end{aligned}\\
&\therefore\ 3n+5\ \text{is even} \\
&\therefore\ \text { a contradiction }\\
&\therefore\ \text { So n cannot be odd }\\
&\therefore \text { n is even. }\end{aligned}$$
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