question about logically true statements?

Question

"Is the statement $(∃xQ(x) ∧ ∃xR(x)) ↔ ∃x(Q(x) ∧ R(x))$ logically true? If it is, explain why. If it isn’t, give an interpretation under which it is false."

because this question exclusively uses ∃x, does that mean we only need to find ONE interpretation for which it is true? so if we defined $R(x)$ $=$ '$x$ is even' and $Q(x)$ $=$ '$x$ is prime' and worked out both sides to be true, that would mean this statement is logically true, correct? if and only if both are true or both are false

so for for $(∃xQ(x) ∧ ∃xR(x))$ we could interpret as "for some integer $x$, $x$ is even and for some integer $x$, $x$ is prime" meaning there exists an integer that is prime and an integer that is even? and the second statement we could say "for some integer $x$, $x$ is prime and even" meaning there exists a number that is both prime and even. so both sides are true, therefore the statement is logically true, correct?

Answer 1

What do you think of "There are cats who are alive and there are cats who are dead" versus "There are cats who are alive and dead" ?

Answer 2

In the universe of natural numbers $\exists x\left[x=4\right]\wedge\exists x\left[x=7\right]$ is a true statement, but $\exists x\left[x=4\wedge x=7\right]$ is not a true statement.