Explanation

In this question, we are given some algebra to work with and we must identify the one or more Roman Numeral statements that "must be true." Whenever we are asking whether something "must be true," we can look for a counterexample (situation in which the statement isn't true); for each Roman Numeral statement, if we find a counterexample, we can eliminate the statement.

Starting with statement I, we can look for a counterexample. Since x is divisible by 2 and 0 < x < 5, x could be something like 2 or 4. In fact, those are the only possible values of x. That means that y is either 4 or 8, so y is definitely an integer. Therefore, statement I must be true. Based on this fact alone, we can eliminate answer choices (B) and (C).

Moving on to statement II, we are still dealing with only two possibilities. Again, either x = 2 and y = 4, or x = 4 and y = 8. We can plug in the values for each case. If x = 2 and y = 4, then statement II would say,

y^{2} > xy

(4)^{2} > (2)(4)

16 > 8

This statement is true. And if x = 4 and y = 8, similarly, we get:

y^{2} > xy

(8)^{2} > (4)(8)

64 > 32

Also a true statement. These are the only two possibilities, and statement II is true in both of them, so statement II must be true.

Since we know that statements I and II both must be true, there is only one possible answer choice, (D), and we don't need to evaluate statement III.

**The correct answer is (D).**

Starting with statement I, we can look for a counterexample. Since x is divisible by 2 and 0 < x < 5, x could be something like 2 or 4. In fact, those are the only possible values of x. That means that y is either 4 or 8, so y is definitely an integer. Therefore, statement I must be true. Based on this fact alone, we can eliminate answer choices (B) and (C).

Moving on to statement II, we are still dealing with only two possibilities. Again, either x = 2 and y = 4, or x = 4 and y = 8. We can plug in the values for each case. If x = 2 and y = 4, then statement II would say,

y

(4)

16 > 8

This statement is true. And if x = 4 and y = 8, similarly, we get:

y

(8)

64 > 32

Also a true statement. These are the only two possibilities, and statement II is true in both of them, so statement II must be true.

Since we know that statements I and II both must be true, there is only one possible answer choice, (D), and we don't need to evaluate statement III.