Answer
There exists a number that is divisible by 4 and not divisible by 8.
Work Step by Step
Recall the definition of necessary:
"$\forall x$, r(x) is a necessary condition for s(x)" means "$\forall x$, if ~r(x) then ~s(x)" or equivalently "$\forall x$, if s(x) then r(x)."
In this case, r(x) is: x is divisible by 8.
s(x) is: x is divisible by 4.
The symbolic form of the necessary statement is: $s(x) \rightarrow r(x)$ which means "if x is divisible by 4, then x is divisible by 8."
Recall the form of the negation of a universal conditional statement:
$~(\forall x, P(x) \rightarrow Q(x)) \equiv \exists x$ such that $(P(x) \land $ ~$Q(x))$
So in this case the negation is: there exists an x such that s(x) and ~r(x) which is, "there exists a number that is divisible by 4 and not divisible by 8."
