$($a$)$ Prove by using the method of contradiction that, if $n$ is an integer and $n^2+3n$ is odd, then $n$ is even.

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$(2$ marks$)$

$($b$)$ Let the sets $P$ and $Q$ have $m$ and $n$ elements respectively. If the cardinality of the power set of $P$ is $112$ more than the cardinality of the power set of $Q,$ find the cardinality of the Cartesian product $PQ.$

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$(3$ marks$)$