7. (i) Given that \(a\) and \(b\) are integers such that $$a + b \text { is odd }$$ Use algebra to prove by contradiction that at least one of \(a\) and \(b\) is odd.
(ii) A student is trying to prove that $$( p + q ) ^ { 2 } < 13 p ^ { 2 } + q ^ { 2 } \quad \text { where } p < 0$$ The student writes: $$\begin{gathered} \qquad \begin{array} { c } p ^ { 2 } + 2 p q + q ^ { 2 } < 13 p ^ { 2 } + q ^ { 2 }
2 p q < 12 p ^ { 2 }
\text { so as } p < 0 \quad 2 q < 12 p
q < 6 p \end{array} \end{gathered}$$ a. Identify the error made in the proof.
b. Write out the correct solution.