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Math Expert V
Joined: 02 Sep 2009
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10 00:00

Difficulty:   95% (hard)

Question Stats: 38% (01:58) correct 62% (02:05) wrong based on 168 sessions

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If $$0 \lt x \lt y$$ and $$x$$ and $$y$$ are consecutive perfect squares, what is the remainder when $$y$$ is divided by $$x$$?

(1) Both $$x$$ and $$y$$ have 3 positive factors.

(2) Both $$\sqrt{x}$$ and $$\sqrt{y}$$ are prime numbers.

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Math Expert V
Joined: 02 Sep 2009
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1
4
Official Solution:

If $$0 \lt x \lt y$$ and $$x$$ and $$y$$ are consecutive perfect squares, what is the remainder when $$y$$ is divided by $$x$$?

Notice that since $$x$$ and $$y$$ are consecutive perfect squares, then $$\sqrt{x}$$ and $$\sqrt{y}$$ are consecutive integers.

(1) Both $$x$$ and $$y$$ have 3 positive factors. This statement implies that $$x=(prime_1)^2$$ and $$y=(prime_2)^2$$. From above we have that $$\sqrt{x}=prime_1$$ and $$\sqrt{y}=prime_2$$ are consecutive integers. The only two consecutive integers which are primes are 2 and 3. Thus, $$x=(prime_1)^2=4$$ and $$y=(prime_2)^2=9$$. The remainder when 9 is divided by 4 is 1. Sufficient.

(2) Both $$\sqrt{x}$$ and $$\sqrt{y}$$ are prime numbers. The same here: $$\sqrt{x}=2$$ and $$\sqrt{y}=3$$. Sufficient.

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##### General Discussion
Intern  B
Joined: 09 Jul 2016
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I think this is a poor-quality question and I don't agree with the explanation. Question statement is not clear enough.
Math Expert V
Joined: 02 Sep 2009
Posts: 60647

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siddhanthsivaraman wrote:
I think this is a poor-quality question and I don't agree with the explanation. This is a poor quality question and meaning of consecutive perfect squares is not clear while attempting...

Consecutive perfect square are say 1^1 = 1 and 2^2 = 4 or 5^2 = 25 and 6^2 = 36.
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poor quality question

From above we have that x√=prime1x=prime1 and y√=prime2y=prime2 are consecutive integers. How was this proved ?
Expect a better explanation.
Math Expert V
Joined: 02 Sep 2009
Posts: 60647

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Yashkumar wrote:
poor quality question

From above we have that x√=prime1x=prime1 and y√=prime2y=prime2 are consecutive integers. How was this proved ?
Expect a better explanation.

x and y are consecutive perfect squares, so x and y could be:
$$x = 1$$ and $$y = 4$$ --> $$\sqrt{x}=1$$ and $$\sqrt{y}=2$$, consecutive integers;
$$x = 4$$ and $$y = 9$$ --> $$\sqrt{x}=2$$ and $$\sqrt{y}=3$$, consecutive integers;
$$x = 9$$ and $$y = 16$$ --> $$\sqrt{x}=3$$ and $$\sqrt{y}=4$$, consecutive integers;
...
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Bunuel wrote:
Yashkumar wrote:
poor quality question

From above we have that x√=prime1x=prime1 and y√=prime2y=prime2 are consecutive integers. How was this proved ?
Expect a better explanation.

x and y are consecutive perfect squares, so x and y could be:
$$x = 1$$ and $$y = 4$$ --> $$\sqrt{x}=1$$ and $$\sqrt{y}=2$$, consecutive integers;
$$x = 4$$ and $$y = 9$$ --> $$\sqrt{x}=2$$ and $$\sqrt{y}=3$$, consecutive integers;
$$x = 9$$ and $$y = 16$$ --> $$\sqrt{x}=3$$ and $$\sqrt{y}=4$$, consecutive integers;
...

Dear Bunuel

As u stated above the consecutive perfect square 1, 4, 9, 16, 25.

From statement 1, it is invalid to consider 9 & 16 because both are consecutive but 16 does not consist of 3 factors. Am I right?
Math Expert V
Joined: 02 Sep 2009
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Mo2men wrote:
Bunuel wrote:
Yashkumar wrote:
poor quality question

From above we have that x√=prime1x=prime1 and y√=prime2y=prime2 are consecutive integers. How was this proved ?
Expect a better explanation.

x and y are consecutive perfect squares, so x and y could be:
$$x = 1$$ and $$y = 4$$ --> $$\sqrt{x}=1$$ and $$\sqrt{y}=2$$, consecutive integers;
$$x = 4$$ and $$y = 9$$ --> $$\sqrt{x}=2$$ and $$\sqrt{y}=3$$, consecutive integers;
$$x = 9$$ and $$y = 16$$ --> $$\sqrt{x}=3$$ and $$\sqrt{y}=4$$, consecutive integers;
...

Dear Bunuel

As u stated above the consecutive perfect square 1, 4, 9, 16, 25.

From statement 1, it is invalid to consider 9 & 16 because both are consecutive but 16 does not consist of 3 factors. Am I right?

We can consider the above values from the stem. From (1), the only values possible are $$x=(prime_1)^2=4$$ and $$y=(prime_2)^2=9$$.
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Bunuel wrote:
Official Solution:

If $$0 \lt x \lt y$$ and $$x$$ and $$y$$ are consecutive perfect squares, what is the remainder when $$y$$ is divided by $$x$$?

Notice that since $$x$$ and $$y$$ are consecutive perfect squares, then $$\sqrt{x}$$ and $$\sqrt{y}$$ are consecutive integers.

(1) Both $$x$$ and $$y$$ have 3 positive factors. This statement implies that $$x=(prime_1)^2$$ and $$y=(prime_2)^2$$. From above we have that $$\sqrt{x}=prime_1$$ and $$\sqrt{y}=prime_2$$ are consecutive integers. The only two consecutive integers which are primes are 2 and 3. Thus, $$x=(prime_1)^2=4$$ and $$y=(prime_2)^2=9$$. The remainder when 9 is divided by 4 is 1. Sufficient.

(2) Both $$\sqrt{x}$$ and $$\sqrt{y}$$ are prime numbers. The same here: $$\sqrt{x}=2$$ and $$\sqrt{y}=3$$. Sufficient.

Hi Bunuel,

In the above question Statement 1 says that both x and y have 3 positive factor and you have written that the statement implies that x=(prime1)2x=(prime1)2 and y=(prime2)2y=(prime2)2. How did you arrive at this? Can you please elaborate? Thanks.
Math Expert V
Joined: 02 Sep 2009
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rohan2345 wrote:
Bunuel wrote:
Official Solution:

If $$0 \lt x \lt y$$ and $$x$$ and $$y$$ are consecutive perfect squares, what is the remainder when $$y$$ is divided by $$x$$?

Notice that since $$x$$ and $$y$$ are consecutive perfect squares, then $$\sqrt{x}$$ and $$\sqrt{y}$$ are consecutive integers.

(1) Both $$x$$ and $$y$$ have 3 positive factors. This statement implies that $$x=(prime_1)^2$$ and $$y=(prime_2)^2$$. From above we have that $$\sqrt{x}=prime_1$$ and $$\sqrt{y}=prime_2$$ are consecutive integers. The only two consecutive integers which are primes are 2 and 3. Thus, $$x=(prime_1)^2=4$$ and $$y=(prime_2)^2=9$$. The remainder when 9 is divided by 4 is 1. Sufficient.

(2) Both $$\sqrt{x}$$ and $$\sqrt{y}$$ are prime numbers. The same here: $$\sqrt{x}=2$$ and $$\sqrt{y}=3$$. Sufficient.

Hi Bunuel,

In the above question Statement 1 says that both x and y have 3 positive factor and you have written that the statement implies that x=(prime1)2x=(prime1)2 and y=(prime2)2y=(prime2)2. How did you arrive at this? Can you please elaborate? Thanks.

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

So, according to the above, an integer to have 3 factor it must be $$x=(prime)^2$$. In this case the number of factors = (2 + 1) = 3: 1, prime and prime^2. For example, 2^2 has three factors 1, 2, and 4; 3^2 has three factors 1, 3, and 9.
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Bunuel wrote:
If $$0 \lt x \lt y$$ and $$x$$ and $$y$$ are consecutive perfect squares, what is the remainder when $$y$$ is divided by $$x$$?

(1) Both $$x$$ and $$y$$ have 3 positive factors.

(2) Both $$\sqrt{x}$$ and $$\sqrt{y}$$ are prime numbers.

D was very prominent , but I overthought ,
1st statement : 3 positive factors , this means the number is prime . But in this case , 5 & 11 can also be a option ? even 5 & 7 ? If y is 7 , and x is 5 then the remainder is 2 , and if y is 3 and x is 2 , then the remainder is 1 . This is the only confusion I'm having .
other that both the statement clearly indicates that x and y are prime .

Statement 2 , clearly says , that x & y are prime , as both are perfect squares

I chose E , as I overlooked that X & Y are consecutive perfect square. I was determined that X & Y both are prime, Now when I'm reading it again , it bloody says consecutive perfect square , hence 2 & 3 are the only option. Makes perfect sense.

Nice question Bunuel .
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Math Expert V
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loserunderachiever wrote:
Bunuel wrote:
If $$0 \lt x \lt y$$ and $$x$$ and $$y$$ are consecutive perfect squares, what is the remainder when $$y$$ is divided by $$x$$?

(1) Both $$x$$ and $$y$$ have 3 positive factors.

(2) Both $$\sqrt{x}$$ and $$\sqrt{y}$$ are prime numbers.

D was very prominent , but I overthought ,
1st statement : 3 positive factors , this means the number is prime . But in this case , 5 & 11 can also be a option ? even 5 & 7 ? If y is 7 , and x is 5 then the remainder is 2 , and if y is 3 and x is 2 , then the remainder is 1 . This is the only confusion I'm having .
other that both the statement clearly indicates that x and y are prime . Can you please explain the remainder keeps on changing if we are considering other primes except 2 and 3 .

I chose E , I was determined that I should be D , but as the remainder was changing I chose E .

Bunuel

It seems that you are missing the crucial part the stem tells us: $$x$$ and $$y$$ are consecutive perfect squares. So, for example:
1^2 and 2^2;
2^2 and 3^2;
3^2 and 4^2;
...

So, if $$x$$ and $$y$$ are consecutive perfect squares, then $$\sqrt{x}$$ and $$\sqrt{y}$$ are consecutive integers:
1 and 2;
2 and 3;
3 and 4;
...

Both statements imply that $$\sqrt{x}$$ and $$\sqrt{y}$$ are primes. The only two consecutive integers which are primes are 2 and 3.
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Manager  B
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Bunuel wrote:
loserunderachiever wrote:
Bunuel wrote:
If $$0 \lt x \lt y$$ and $$x$$ and $$y$$ are consecutive perfect squares, what is the remainder when $$y$$ is divided by $$x$$?

(1) Both $$x$$ and $$y$$ have 3 positive factors.

(2) Both $$\sqrt{x}$$ and $$\sqrt{y}$$ are prime numbers.

D was very prominent , but I overthought ,
1st statement : 3 positive factors , this means the number is prime . But in this case , 5 & 11 can also be a option ? even 5 & 7 ? If y is 7 , and x is 5 then the remainder is 2 , and if y is 3 and x is 2 , then the remainder is 1 . This is the only confusion I'm having .
other that both the statement clearly indicates that x and y are prime . Can you please explain the remainder keeps on changing if we are considering other primes except 2 and 3 .

I chose E , I was determined that I should be D , but as the remainder was changing I chose E .

Bunuel

It seems that you are missing the crucial part the stem tells us: $$x$$ and $$y$$ are consecutive perfect squares. So, for example:
1^2 and 2^2;
2^2 and 3^2;
3^2 and 4^2;
...

So, if $$x$$ and $$y$$ are consecutive perfect squares, then $$\sqrt{x}$$ and $$\sqrt{y}$$ are consecutive integers:
1 and 2;
2 and 3;
3 and 4;
...

Both statements imply that $$\sqrt{x}$$ and $$\sqrt{y}$$ are primes. The only two consecutive integers which are primes are 2 and 3.

Makes sense ! bloodyhell , I missed the question stem as consecutive perfect squares. Thank you Bunuel . Nice question.
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I think this is a high-quality question and I don't agree with the explanation. Hi Bunuel,
I went through the explanation and the other comments.
I may be wrong but below is the reason I think this needs more discussion.
All squares of prime numbers have exactly 3 factors. However the question stem states that the Squares of primes are consecutive. Not that the Prime numbers themselves are consecutive. So we can have any primes picked. 2,3 or 5,7 or anything like 11,13. In all these cases rootX= the prime1 and rootY is the Prime2. But if you see all the different X and Y for these pair of primes will yield different Remainders. This is precisely the reason I think Statement 1 is insufficient. Try 169(y)/121(x) = 48(r). 49(y)/25(x)=24(r)..
In that same logic Statement 1 & 2 combined doesn't help getting a fixed answer. Thus I went for choice E.
Kindly help me understand If I read something wrong.
Math Expert V
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saplogics wrote:
I think this is a high-quality question and I don't agree with the explanation. Hi Bunuel,
I went through the explanation and the other comments.
I may be wrong but below is the reason I think this needs more discussion.
All squares of prime numbers have exactly 3 factors. However the question stem states that the Squares of primes are consecutive. Not that the Prime numbers themselves are consecutive. So we can have any primes picked. 2,3 or 5,7 or anything like 11,13. In all these cases rootX= the prime1 and rootY is the Prime2. But if you see all the different X and Y for these pair of primes will yield different Remainders. This is precisely the reason I think Statement 1 is insufficient. Try 169(y)/121(x) = 48(r). 49(y)/25(x)=24(r)..
In that same logic Statement 1 & 2 combined doesn't help getting a fixed answer. Thus I went for choice E.
Kindly help me understand If I read something wrong.

169 =13^2 and 121 = 11^2 are not consecutive perfect square. 169 = 13^2 and 144 = 12^2 are. Or 144 = 12^2 and 11^2.
49 =7^2 and 121 = 5^2 are not consecutive perfect square.
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Got it. Thanks for Opening my eyes. Thank you so much. Manager  B
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Bunuel...
Reg Statement (2)..shouldnt we need to consider that Root(X) and Root(y) can also be the negative numbers...thus rendering the Statement insufficient?
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Debashis Roy wrote:
Bunuel...
Reg Statement (2)..shouldnt we need to consider that Root(X) and Root(y) can also be the negative numbers...thus rendering the Statement insufficient?

First of all, (2) says that $$\sqrt{x}$$ and $$\sqrt{y}$$ are prime numbers. Only positive numbers can be primes.

Next, when the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root. That is:

$$\sqrt{9} = 3$$, NOT +3 or -3;
$$\sqrt{16} = 2$$, NOT +2 or -2;

Notice that in contrast, the equation $$x^2 = 9$$ has TWO solutions, +3 and -3. Because $$x^2 = 9$$ means that $$x =-\sqrt{9}=-3$$ or $$x=\sqrt{9}=3$$.
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I think this is a poor-quality question and I don't agree with the explanation. x=16,y=25 satisfies the stem and (1)

why do we need to restrict x and y to 2,3?
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diksav wrote:
I think this is a poor-quality question and I don't agree with the explanation. x=16,y=25 satisfies the stem and (1)

why do we need to restrict x and y to 2,3?

(1) says that Both x and y have 3 positive factors. Does 16 have 3 positive factors? Please re-read the discussion.
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