If x, y and z are distinct prime numbers, how many positive factors do

nice question !!

its A..

(z+1) (z+1)

As with many GMAT Quant questions, this problem is easier than it looks if you know how to identify the leverage embedded in the problem. It's not just about the math. Remember: while the GMAT might structure its problems around Quant and Verbal principles, the GMAT is primarily a critical-thinking test. So, for those of you studying for the GMAT, you will want to internalize solid strategies that help you to identify and solve entire classes of questions. Pattern recognition is key.

In this case, the words "distinct" and "prime" drastically limit the number of possibilities. We can use this to our advantage. If a number is "distinct", this means it is unique from all the other possible numbers. A "prime" number has only two positive factors: \(1\) (which is a factor of every integer regardless of the number) and itself.

The target of the question also makes this problem easier. We are only asked to figure out "how many positive factors does \((xy)^z\) have?" (instead of "what are the positive factors \((xy)^z\)?".) This is a large distinction. After all, prime numbers are incredibly limited in the number of factors they can have! (Incidentally, for those that want an equation: for \(N=x^a∗y^b∗z^c\), where \(x\), \(y\), and \(z\) are distinct prime factors of \(N\), the number of factors of \(N\) will be calculated by the expression \((a+1)(b+1)(c+1)\).)

Statement #1 tells us that \(z=5\). While we don't know what \(x\) or \(y\) is, we know they are "distinct primes", which means that \(xy\) can only have \(4\) factors: \(1\), \(x\), \(y\), and \(xy\). But if we know this, then we also know that multiplying \(xy\) by itself \(5\) times will also result in a specific, finite number of factor combinations. Of course, we can count them out. Of course, we can plug them into an equation. But this is Data Sufficiency. "Minimize Your Math." As soon as we realize the information given us will result in one -- and only one -- answer, we can stop doing the problem. We don't even need to know the ANSWER to the question stem. We just need to know that there will be only one answer. Since we know the math WILL resolve to a single answer, we don't need to calculate what that answer is. Statement #1 is sufficient.

Statement #2 can be broken by a strategy I call "Easy Numbers." If you can invent some easy numbers that follow the rules of the problem but that give you different answers to the question, then the statement is insufficient. Statement #2 says that "\(x+y = 10\)." One possibility for \(x\) and \(y\) would be \(x=3\) and \(y=7\), making \(xy=21\). But if we don't know how many times we would be multiplying \(21\) by itself (in other words, if we don't know \(z\)), there would be no way to conclusively determine the number of possible factors. \(21^z\) could be infinitely large with an infinite number of factors. We can stop as soon as we realize that there are two or more options. Statement #2 is insufficient.

Since Statement #1 is sufficient and Statement #2 is insufficient, the answer is A.

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