This is a Yes/No question, so either a definite “yes” or a definite “no” would be Sufficient. We are told that x – 1 is an integer greater than 0, so x must be an integer greater than 1. We need to know whether the expression x – 1 is definitely divisible by 3, which means that when x — 1 is divided by 3, the result is an integer. This will be so if and only if x – 1 is a multiple of 3.

Evaluate the Statements:

Statement (1): While we could FOIL (\(x2\) + x)(x + 1), doing so would produce something even more complicated. It’s simpler to factor out the x common to both terms in (x2 + x):

(\(x2\) + x)(x + 1)

x(x + 1)(x + 1)

Statement (1) tells us that x(x + 1)(x + 1) is divisible by 3, which means that either x or x + 1 has to be a multiple of 3.

Multiples of 3 only come along every third integer. So if x is a multiple of 3, then x – 1 definitely can’t be (x– 3 would be, but not x – 1). Similarly, if x + 1 is a multiple of 3, then again x – 1 definitely can’t be (x – 2 would be, but not x – 1). Since x – 1 is definitely not a multiple of 3, this Statement is Sufficient.

Picking Numbers can help us to illustrate this. If we let x = 3, then x – 1 = 2, which answers our question with a “no”; 2 is not divisible by 3. If we pick x + 1 = 3, then x – 1 = 1. This also gives us a “no,” as 1 is not divisible by 3.

With always producing a “no,” this Statement is Sufficient to answer the question, and we can eliminate choices (B), (C), and (E).

Statement (2): As before, we can simplify the given expression by factoring:

x(\(x2\)–x)

x(x)(x– 1)

Using the same reasoning as before, we know that if this expression is divisible by 3, then either x or x – 1 is divisible by 3. If x is divisible by 3, then x – 1 will not be. In other words, the expression x – 1 may or may not be divisible by 3. This statement is therefore Insufficient to answer the question. Eliminate choice (D).

Therefore, Answer Choice (A) is correct.

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