If z = x*y^3, where x and y are positive integers, what is the value

Question

If z = x*y^3, where x and y are positive integers, what is the value of y ?

(1) The number of different positive factors of z is 4.
(2) z is the product of two different prime numbers.

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gmatophobia · Answer

Given

z = x * y^3

x \geq{1}  ;  y \geq{1}

Question

y = ?

Statement 1

The number of different positive factors of z is 4.

z = x * y^3

Now y can be = 1, and x can be a cube of a prime number.

So  x = p^3

z = p^3 * 1

Alternatively,

y can be any prime number and x = 1

Hence, we can  eliminate A and D

Statement 2

z is the product of two different prime numbers.

So,  z = p_1 * p_2

Also   z = x * y^3

Now there can be multiple scenarios we can think of

y is a prime number and x is also a prime number : In that case, z is a multiple of more than two prime numbers, hence this is not a valid case.
 
  y is a prime number and x = 1 : In that case, z is again a multiple of more than two prime numbers, hence this is also not a valid case.
 
   x = p_1 * p_2  and y = 1 : This is valid.

Note : We don't even have to consider scenario 1 and scenario 2 (to save time). It's given that y is an integer, so if y has any prime factors in it, z will be a product of more than two prime factors. Hence y has to be equal to 1.

Thus have a definite answer.

IMO B

Archit3110 · Answer

given
z = x*y^3, where x and y are positive integers
target value of y?
#1
 The number of different positive factors of z is 4.
z=x*y^3
case 1 ; x is 1 and y is prime number 
case 2 : x is cube of prime number and y is 1
insufficient
#2
z is the product of two different prime numbers
least value of z is 6 
z = x*y^3
x has to be 6 and y is 1
sufficient
option B

SaquibHGMATWhiz · Answer

Solution: (Methodical)

Pre Analysis:  
  We are given  z = x*y^3  where x and y are positive integers. This means variable z will also be a positive integer  
 We are asked the value of y which is  y=\sqrt {\frac{z}{x}}  
 Since y is a positive integer, z has to be a cube multiple of x like  z=x ,  z=8x ,  z=27x  etc

Statement 1:   The number of different positive factors of z is 4.
  According to this statement, either  z=p^3  where p is a prime number or  z=p_1	imes p_2  where  p_1  and  p_2  are teo distinct prime numbers  
 If  z=p^3 , then  y=\sqrt {\frac{z}{x}}=\sqrt {\frac{p^3}{x}}=\frac{p}{\sqrt {x}}   In this case,  y=p  when  x=1  or  y=1  when  x=p^3    
 If  z=p_1	imes p_2 , then  y=\sqrt {\frac{z}{x}}=\sqrt {\frac{p_1	imes p_2}{x}}    In this case,  y=1  when  x=p_1	imes p_2  is the only option    
 Thus,  statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2:   z is the product of two different prime numbers
  This is the second case of statement 1 where  y=\sqrt {\frac{z}{x}}=\sqrt {\frac{p_1	imes p_2}{x}}    In this case,  y=1  when  x=p_1	imes p_2  is the only option   
 Thus,  statement 2 alone is sufficient

Hence the right answer is    Option B

Solution: (Logical)

Statement 1: If  z = x*y^3  and z has 4 distinct factors then  one of x or y has to be 1. Both of them cannot be greater than 1    Otherwise z will not have 4 factors but more   
 Statement 2: If  z = x*y^3  and z is the product of two different prime numbers, then  y has to be 1

Hence the right answer is    Option B

Note: We are currently working on an article in which we will try and cover the above concept in detail. Stay tuned.

shubhim20 · Answer

Bunuel  the second can't have 54 as 2*3 to power 3 or does it have to be jjust product of two prime that is 6 or wwhy can't it be 10?

Bunuel · Answer

From (2), x can be the product of any two different primes, not necessarily 2 and 3. The key point is that in every such case, y must be 1, which is exactly what the question asks for.

I remember telling you this but here it is again:

Pure algebraic questions are no longer a part of the  DS syllabus  of the GMAT.
 
DS questions in GMAT Focus encompass various types of word problems, such as:

Word Problems 
 Work Problems 
 Distance Problems 
 Mixture Problems 
 Percent and Interest Problems 
 Overlapping Sets Problems 
 Statistics Problems 
 Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check  GMAT Syllabus for Focus Edition

You can also visit the  Data Sufficiency forum  and filter questions by  OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources  to see the types of questions currently tested on the GMAT.

So, you can ignore this question. 
 
Hope it helps.

If z = x*y^3, where x and y are positive integers, what is the value

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GMAT Data Sufficiency (DS) Questions

If z = x*y^3, where x and y are positive integers, what is the value

Prep Toolkit

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