Direct proof
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Use direct proof to prove that the sum of any two odd integers is even.
$$\begin{aligned}
&\text{let p and q be odd integers}\\
&\text{Hence}\ p=2 k+1\ \text{for some integer}\ k\\
&\text{and}\ q=2j+1\ \text{for some integer}\ j.\\
&\begin{aligned}
p+9 &=2 k+1+2 s+1 \\
&=2(k+1+1) \\
&=2 m \text { were } m=k+j+1\ \text { (an integer) }
\end{aligned}\\
&\therefore\ p+q\ \text{is even.}
\end{aligned}
$$
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