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

Question

If x, y and z are distinct prime numbers, how many positive factors does (xy)^z have?

(1) z = 5

(2) x + y = 10

abhishekkpv · Answer

A for me

Given that x,y and z are three distinct prime numbers 
frm statement 1) z=5

so (xy)^5 = (x^5)*(y^5)
and so the no of positive factors is (5+1)*(5+1)=6*6=36

frm statement 2) x+y = 10
which means the possible combination is 3,7 or 7,3
if z=1 we have 4 factors
if z=2 we will have 9 factors and so on

so 2 is insufficient

Bunuel · Answer

If x, y and z are distinct prime numbers, how many positive factors does (xy)^z have?

\((xy)^z=x^z*y^z\). Now, since x and y are distinct prime numbers, then the number of factors equal to (z+1)(z+1) (check here: https://gmatclub.com/forum/math-number-theory-88376.html). Thus, all we need to know to answer the question is the value of z.

(1) z=5. Sufficient.

(2) x+y = 10. Not sufficient.

Answer: A.

sanjoo · Answer

nice question !!

its A..

(z+1) (z+1)

AaronPond · Answer

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  .

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards