Convert the following statements to propositions. Use appropriate quantifiers and make sure all variables are bound:

There exists a number that is divisible by both $2$ and $3$

For every hobby, there is someone who enjoys it$.$

There are exactly two people who brought cake to the party

Show that this argument is valid without a truth table

$($be sure to number each step and justify$)$:

$pq$

$p^{{?}^{?}}r$

:$.q^{{?}^{?}}r$