If x, y and z are positive integers, and y and z are factors of x^2

Question

Competition Mode Question

If x, y and z are positive integers, and y and z are factors of x^2, is x divisible by y?

(1) y is a prime number.
(2) z = y^2

shameekv1989 · Answer

If x, y and z are positive integers, and y and z are factors of  x^2 , is x divisible by y?

(1) y is a prime number.
(2)  z = y^2

Answer - D

Given  x,y,z > 0  and are integers

\frac{x^2}{y}= Integer  &  \frac{x^2}{z}= Integer

Property tested :-  x^2  shares the same unique prime factors as x

Statement 1) y is a prime number i.e. it is not an exponent of any number -> this prime number is also in  x^2  (since  \frac{x^2}{y } is an integer) -> y prime number is also in x ->  \frac{x}{y}  is an integer

For eg y = 13; x will be anything that will have  13^2  min i.e. x = 13 or 13*2 or 13*3 ....

Statement 2)  z = y^2 
since  \frac{x^2 }{ z } = integer ->  \frac{x^2}{y^2 } = Integer ->  (\frac{x}{y})^ 2  = integer -> RHS (integer) is a perfect square ->  \frac{x}{y}  is an integer (Since if  \frac{x}{y}  were not an integer then  \frac{x^2}{y ^ 2}  would not be an integer).

Answer - D

Asalways · Answer

1) if y is a prime factor of x^2, then x is divisible by y
Suff

2) if z=y^2, then y must be a prime factor of x^2
Suff

Answer: D

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GMATWhizTeam · Answer

Solution:
  y is a factor of  x^2 
   \frac{x^2}{y}=k,  where k is an integer.
  x =√(k*y)  
 z is a factor of x^2
   \frac{x^2}{z}  is an integer.
  x =√(M*z) , where M is an integer  
Statement 1: y is a prime number
   x =√(k*y) , where y is a prime number.
 It means k must be a multiple of y and some perfect square, otherwise, x will not be an integer.
 Let,  k =y*N^2 , where N is an integer.
  x =y*N \
 \frac{x}{y} =N 
 y is a factor of x 
Hence, statement 1 is sufficient, we can eliminate answer options B, C, and E.
Statement 2:
   x =√(M*z)
 x =√(M*y^2)
 x =y√M,  here M must be a perfect square otherwise, x will not be an integer.
 Let  M = Q^2 , where Q is an integer.
  \frac{x}{y} = Q 
 y is a factor of x. 
 Hence, statement 2 is also sufficient, the correct answer is Option D.

CareerGeek · Answer

y  and  z  are factors of  x^2 
-->  \frac{x^2}{y} = a  &  \frac{x^2}{z} = b  for some positive integers  a  &  b

(1)  y  is a prime number.
 \frac{x^2}{y} = a 
-->  x^2 = a*y 
Since  x  &  y  are positive integers, " a " Will definitely be of the form  y*d^2  for some integer  d 
-->  x^2 = (y*d^2)*y = (d*y)^2 
-->  x = yd 
-->  x  is divisible by  y  -->  Sufficient

(2)  z = y^2 
 z  is a factor of  x^2 
-->  y^2  is a factor of  x^2 
--> Since  x  &  y  are positive integers,  y  will definitely be a factor of  x  -->  Sufficient

Option D

joohwangie · Answer

If x, y and z are positive integers, and y and z are factors of x^2, is x divisible by y?

x^2 is divisible by both y and z

(1) y is a prime number.
if y is prime then x^2 must 
if x^2 is 25 and y is 5 z is 25 x is 5
yes
if x^2 is 81 and y is 3 z is 9 x is 9
yes
therefore
sufficient

(2) z = y^2
if z=25 y=5 then x=5
sufficient too

D

hardikajmani · Answer

n  and  p  are factors of   m^2

for some positive integers  a & b 
 \frac{m^2 }{ n }= a  and  \frac{m^2}{p} = b

1)  n  is a prime number
 \frac{m^2 }{ n } = a 
 m^2 = a * n 
Since  m & n   are positive integers,  "a"  will be in the form of  n * d^2 
 m^2 = (n * d^2) * n = (d * y)^2 
 m = n * d 
Hence  x  is divisible by  y .

2)  p = n^2 
 p  is a factor of  n^2 
 n^2  is a factor  m^2 
Since  m & n  are positive integers,  n  will definitely be a factor of  m .

Option D

Archit3110 · Answer

given 
x^2 = z*y
#1
y is a prime number
since z & y are factors of x^2 ; so when x is divided by y its factor would be z ; yes
#2
z = y^2
or say y^3 = x^2
only possible when x=y=1
sufficient
IMO D

If x, y and z are positive integers, and y and z are factors of x^2, is x divisible by y?

(1) y is a prime number.
(2) z = y^2

exc4libur · Answer

x,y,z are positive integers

f(xˆ2)=y,z

Is x/y=integer?

(1) y is a prime number.   sufic

if y is a prime of xˆ2 then it is also a prime factor of x

(2) z = y^2   sufic

z=yˆ2, then yˆ3 is a factor of xˆ2;
so, y must be a factor x.

Ans (D)

ostrick5465 · Answer

(1) y is a prime number 
=> y is a factor of x^2 also y is a factor of x
=> x divisible by y
=> SUFF

(2) z = y^2
z is a factor of x^2 
=> y^2 is a factor of x^2 
=> y is a factor of x
=> x divisible by y
=> SUFF

=>  Choice D

Jawad001 · Answer

If x, y and z are positive integers, and y and z are factors of x^2, is x divisible by y?

(1) y is a prime number.
(2) z = y^2

We are given that, y and z are factors of x^2.
Let x = 3
x^2 = 3^2= 9 = 9,3,1
From statement (1), y = prime number = 3, 
 z = 9 and 
 x = 3
In this case, x is divisible by y.

In another way, Let x = 12,
x^2 = (12)^2= 144= 144,72, 48,36,24,18,12,8………..3,2,1
x = 12, and y = 3 or 2

In this case also, In this case, x is divisible by y. 
From statement (2), For the above two cases, x divisible by y when z = y^2
Answer:D

freedom128 · Answer

y and z are either 1,x, or x^2 or a factor of x

(1) y is a prime factor of x. So x can definitely be divided by y. 
SUFFICIENT

(2) z = y^2
x=6, x^2=36, y=2, z=4 shows that x can be divided by y.
x=2, x^2=4, y=2, z=4 shows that x can be divided by y.
SUFFICIENT

FINAL ANSWER IS (D)

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minustark · Answer

If x, y and z are positive integers, and y and z are factors of x^2, is x divisible by y?

(1) y is a prime number.
(2) z = y^2

from question stem, y and z are factors of x^2. So when x = 4, y and z can be 1, 2, 4, 8, 16. here when y = 1, 2, 4, x is divisible by y but not for the other two values of y. when x =5, y and z can be 1, 5, 25. when x= 6, y and z can be 1, 2, 3, 4, 6, 9, 12, 18, 36.

1) since y is a prime number, so x^2 will be the square of the same prime number and x will be the multiple of that prime number. so in any case x is divisible by y. sufficient.

2) from this information, we cannot determine any information about the relationship of x and y. so not sufficient.

A is the answer.

lacktutor · Answer

If x, y and z are positive integers, and y and z are factors of  x^2 , is x divisible by y?

( Statement1 ): y is a prime number.
Since y is a prime and a factor of  x^{2} , y is a factor of x too.
 Sufficient

( Statement2 ): z =  y^2 
--> Since  y  and  y^{2}  is a factor of  x^{2} , x is definitely divisible by y 
 Sufficient
 
 The answer is D.

eakabuah · Answer

We are informed that x,y, and z are positive integers and that y and z are factors of x^2. We are to determine if y is a factor of x.

Statement 1: y is a prime number.
This is sufficient. Because knowing that y and are factors of x^2 means that x^2=(y^a * z^b)*m, where a,b, and m are positive integers.
Statement 1 means that a=1 since y is prime. In addition, y cannot be gotten from the square of any other integer or the product of any two integers, hence if y is a factor of x^2, then y must also be a factor of x.

Statement 2: z=y^2
Sufficient. We know that x^2 =(y^a * z^b)*m, where a,b, and m are integers. For the purpose of statement 2, we can assume that that a=b=1.
Then x^2 = yzm, and since z=y^2, then x^2=y^3 * m
x=y√(ym) implying that y is a factor of x.

The answer is D.

bumpbot · Answer

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If x, y and z are positive integers, and y and z are factors of x^2

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