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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}

Re: Consecutive Perfect Sqaures [#permalink]
31 Jan 2012, 16:41

20

This post received KUDOS

Expert's post

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This post was BOOKMARKED

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.

Re: If x, y, and z are positive integers, where x > y and z = √x [#permalink]
10 Mar 2012, 02:13

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. _________________

If you like it, Kudo it!

"There is no alternative to hard work. If you don't do it now, you'll probably have to do it later. If you didn't need it now, you probably did it earlier. But there is no escaping it."

Re: If x, y, and z are positive integers, where x > y and z = √x [#permalink]
12 Jun 2013, 22:33

3

This post received KUDOS

enigma123 wrote:

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

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. _________________

If you found my post useful and/or interesting - you are welcome to give kudos!

Re: If x, y, and z are positive integers, where x > y and z = √x [#permalink]
03 Sep 2013, 09:36

ziko wrote:

enigma123 wrote:

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

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.

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,

Re: If x, y, and z are positive integers, where x > y and z = √x [#permalink]
27 Oct 2014, 07:32

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