If x, y, and z are positive integers, where x y and z = √x

Question

If x, y, and z are positive integers, where x > y and z = √x , are x and y consecutive perfect squares?

(1) x + y = 8z +1 
(2) x – y = 2z – 1

OA is B. But I got stuck after a while. Please see below my solution.

From question stem

z = \sqrt{x}
i.e. x =  z^2  ---------------------------------------------(1)

Considering Statement 1

y = 8z-x+1 -----------------------------------------------------(2)

Putting some values of x and z

When x =2 , z =4

Substituting these values in Equation 2 we will get y = 13. But this cannot be applied as x >y. So I tried another values which are

x = 25 then z =5

Substituting these values in Equation 2 we will get y = 16. Therefore , YES x and y are perfect squares.

I tried one more value which was

when x = 36, z =6

Substituting these values in Equation 2 we will get y = 13. Therefore , NO x and y are NOT perfect squares. 
Two conclusions, therefore this statement alone is NOT sufficient.

Considering Statement 2

After simplifying we will get y =  (z-1)^2 
Therefore z-1 = \sqrt{y}

But from statement 1 we can say z = \sqrt{x}

Therefore, \sqrt{x} - 1 = \sqrt{y}

\sqrt{x} = \sqrt{y} + 1

I am stuck after this. Can someone please help?

Bunuel · Accepted Answer

If x, y, and z are positive integers, where x > y and z = √x , are x and y consecutive perfect squares?

If  x  and  y  are consecutive perfect squares, then  \sqrt{y}  and  \sqrt{x}  must be consecutive integers. So, the question becomes: is  \sqrt{y}=\sqrt{x}-1 ? --> square both sides: is  y=x-2\sqrt{x}+1 ?

(1) x + y = 8z +1 -->  x+y=8\sqrt{x}+1  -->  y=8\sqrt{x}+1-x . Now, question becomes is  8\sqrt{x}+1-x=x-2\sqrt{x}+1 ? --> is  10\sqrt{x}=2x ? --> is  5\sqrt{x}=x ? --> is  \sqrt{x}=5 ? --> is  x=25 ? But we don't know that, hence insufficient.

(2) x – y = 2z – 1 -->  y=x-2\sqrt{x}+1 , which is exactly what we needed to know. Sufficient.

Answer: B.

Hope it's clear.

budablasta · Answer

Isn't this question a good candidate for plugging in simple numbers? I plugged in x=4 and y=1 and got the answer pretty quickly.

Bunuel · Answer

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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ziko · Answer

Excellent explanation by Bunuel, but i will try to give some alternative solution. Since the question askes whether x and y are consecutive perfect squares, if yes they must be perfect numbers in all cases, so we can plug some numbers and check. While plugging in our task is to find numbers when x and y are consecutive and when they are not.

Lets take 4, 9, 16 and 25 - consecutive perfect squares.

(1) if x=9, y=4, z=3, then according to the statement we have 9+4=24+1 which is not true, in case of x=16, y=9 and z=4 we have 16-9=8-1 which is again true. Lets try x=25, y=16, z=5 so we have 25-16=10-1, true again. So in all cases have the same conclusions based on the statement. Sufficient - B.

Hope that helps.

(2) if x=9, y=4, z=3, then according to the statement we have 9-4=6-1 which is true, in case of x=16, y=9 and z=4 we have 16-9=+1 which is again not true. When plug the numbers it is advisable to check at least 3 numbers. So lets try x=25, y=16, z=5 so we have 25+16=40+1 - Bingo! In this case the statement supports that X and Y are consecutive. So we have two different conclusions based on the statement. Not sufficient.

stne · Answer

Solution is truly great but the way it has been explained is confusing !Also there are typos !

in the highlighted part above we are testing statement 2 so 16-9=8-1 = 7 which is true !

More simply :

1) x + y = 8z +1

Work backwards take values of Z find x, and then find y. (  x= z^{2}\hs{3}  put x and z in statement and find y )

If z=4 ,x= 16 we get y = 17 cannot take this case as condition is that x>y ( for all positive z<5, y>x , hence we cannot take those cases )
If z=5, x=25,y=16, does satisfy, here x and y  are  consecutive perfect squares 
If z=6,x=36, y=13,  also does satisfy,  here x and y are  not  consecutive perfect squares.In fact y is not even a perfect square.
If z=7, x=49,y=9  also does satisfy,  here x and y are  not  consecutive perfect squares.

Two cases hence insufficient.

(2) x – y = 2z – 1

Again work backwards take values of Z find x, and then find y. (  x= z^{2}\hs{3}  put x and z in statement and find y )

if Z=1 then x=1 on solving we get y =0  cannot take  this case as y is supposed to be positive.
 
 if Z= 2 then x=4 on solving and finding y we get y =1 , here 4 and 1 are consecutive perfect squares 
 if Z=3 then x=9 then y = 4 , here also x and y are consecutive perfect squares 
 if z=4 then x= 16 then y = 9 also satisfies the equation and x and y are perfect squares 
 . ........
 Take any positive integer for z and find x and y we will see both x and y are consecutive perfect squares
 Lets take z= 10 then x= 100 and we get y =81 here also x and y are consecutive perfect squares,

hence B is sufficient.

Hope this helps.

davesinger786 · Answer

Can we just consider the consecutive perfect squares as  n^2  and  (n-1)^2  and the difference could then be expressed as  (n^2) -  (n^2-2n+1) which is the same as 2n-1 which indicates that Statement 2 is sufficient?

zanaik89 · Answer

Hi Bunuel

How did u get? 5 x=x---- x=5?

Pls explain thanks

Bunuel · Answer

Reduce by  \sqrt{x} :

5\sqrt{x}=x ;

5\sqrt{x}=(\sqrt{x})^2 ;

5=\sqrt{x} .

j.shivank · Answer

Hi Bunuel,

How is \sqrt{x}=5 not sufficient?

Bunuel · Answer

We did not get that  \sqrt{x}=5 ,the question became: is  \sqrt{x}=5 ?

GMATisLovE · Answer

It is given that x and y are 2 consecutive squired. This means they are squires of 2 integers whose mod will give 2 consecutive integers.

Let

x =  n^2 
y =  (n-1)^2  As x > y

Option 1

x+y =8z+1 =>  n^2  +  (n-1)^2  = 8n+1 => 2 n^2  - 2n +1 = 8n +1 =>  2n^2  - 10n=0 => n=0,5 
The equation satisfied for 2 values of n, while for other values of n it do not satisfy.
Hence, INSUFFICIENT

Option 2

x-y = 2n-1 =>  n^2  -  (n-1)^2  = 2n-1 =>  n^2  - ( n^2  +1 -2n) = 2n -1 => 2n -1 = 2n -1
Both LHS and RHS are equal. The equation will satisfy for all values of n.
Hence, SUFFICIENT

Answer "B"

Shobhit7 · Answer

In order to determine that x and y are consecutive squares, we need to plug in y=(z-1)^2 with x=z^2 as given. If, in an equation, LHS=RHS, then its Correct.

St.1: x+y=8z+1
z^2+(z-1)^2=8z+1
z^2+z^2+1-2z=8z+1
2z^2-10z=0
Therefore, for LHS=RHS, Z must be 5. (Z cannot be 0, being a positive interger).We are not given value of Z.
Hence, Insufficient

St.2: x-y=2z-1
z^2-(z-1)^2=2z-1
z^2-(z^2+1-2z)=2z-1
2z-1=2z-1
LHS=RHS
Sufficient.

Hence, Ans B

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rahul16singh28 · Answer

Bumping for an alternative approach -

Statement I: This statement can be written as

z = \sqrt{17-y} + 4 
 When y = 1, x = 64, z = 8 ---> X,Y Not Consecutive Squares
When y = 16, x = 25, z = 5 --> x, y Consecutive squares.

Statement II: This Statement can be written as

y = (z-1)^2  ---->  X, Y are Consecutive Squares.

Hence, B.

kagrawal16 · Answer

Hi  Bunuel ,

Can you help understand in this solution how we deduce Root(y) is an integer.

We start with "If x and y are consecutive squares then Root(x)-Root(y) = 1". We say x and y are consecutive squares to begin with and therefore we form the equation and check with Options 1 and 2.

We know from the question that z=Root(x) and therefore Root(x) is an integer. However for Root(y) we have no such information is given.

Thanks

bumpbot · Answer

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If x, y, and z are positive integers, where x > y and z = √x

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