Perfect Squares

[hja379]

If x and y are positive integers, what is the value of x ?
(1) x, y, and xy are distinct perfect squares less than 100.
(2) x<5

OA is C

[gig92]

If x and y are positive integers, what is the value of x ?
(1) x, y, and xy are distinct perfect squares less than 100.
(2) x<5

OA after discussion.

Given x & y > 0

Stmnt 2 : x < 5 -- Insuff

Stmnt 1 : x, y, and xy perfect square less than 100

as such there can be many values of x and y

x y
1 9
4 9
4 16
16 4 Thus Insuff

From Stmnt 1 and 2

x < 5
xy < 100

Now, only possible values of x, y:

x y
1 4
4 4
1 1

so Ans, IMO:
E

[kmittal82]

(E) in my opinion, here's why:

(1)
Perfect square between 0 and 100 = 1,4,9,16,25,36,49,64,81

If we set either x or y as 1, then the other could be either value and satisfy the "xy" is a perfect square argument
If we set either as 4, then the other number can be 1,9, or16, since no other combination would satisfy the "xy" requirement

So, insufficient

(2) x < 5 could mean x = 1,2,3 or 4, not sufficient in itself

Combine (1) and (2)
The possible value for x could be 1 or 4, so still not sufficient.

[GMATGuruNY]

If x and y are positive integers, what is the value of x ?
(1) x, y, and xy are distinct perfect squares less than 100.
(2) x<5

OA after discussion.

Statement 1: x, y and xy are distinct perfect squares less than 100.
x=1 is not possible because then xy = y, which does not satisfy the condition that y and xy be different values.
y=1 is not possible because then xy = x, which does not satisfy the condition that x and xy be different values.
The following combinations work:
x=4, y=9, xy = 4*9 = 36.
x=9, y=4, xy = 9*4 = 36.

Since x=4 and x=9 each satisfy statement 1, insufficient.

Statement 2: x<5.
x could be any integer between 0 and 5.
Insufficient.

Statements 1 and 2 together:
Since x=1 does not satisfy statement 1, and x=4 is the only other perfect square less than 5, we know that x=4.
Sufficient.

The correct answer is C.

[kmittal82]

If x and y are positive integers, what is the value of x ?
(1) x, y, and xy are distinct perfect squares less than 100.
(2) x<5

OA after discussion.

Statement 1: x, y and xy are distinct perfect squares less than 100.
x=1 is not possible because then xy = y, which does not satisfy the condition that y and xy be different values.
y=1 is not possible because then xy = x, which does not satisfy the condition that x and xy be different values.
The following combinations work:
x=4, y=9, xy = 4*9 = 36.
x=9, y=4, xy = 9*4 = 36.

Since x=4 and x=9 each satisfy statement 1, insufficient.

Statement 2: x<5.
x could be any integer between 0 and 5.
Insufficient.

Statements 1 and 2 together:
Since x=1 does not satisfy statement 1, and x=4 is the only other perfect square less than 5, we know that x=4.
Sufficient.

The correct answer is C.

>x=1 is not possible because then xy = y, which does not satisfy the condition that y and xy be different values.
y=1 is not possible because then xy = x, which does not satisfy the condition that x and xy be different values.

D'oh!! completely missed that point... good question

[hja379]

Tricky one. I updated the OA.

@Mitch- Thanks for the response.

[gig92]

If x and y are positive integers, what is the value of x ?
(1) x, y, and xy are distinct perfect squares less than 100.
(2) x<5

OA after discussion.

Statement 1: x, y and xy are distinct perfect squares less than 100.
x=1 is not possible because then xy = y, which does not satisfy the condition that y and xy be different values.
y=1 is not possible because then xy = x, which does not satisfy the condition that x and xy be different values.
The following combinations work:
x=4, y=9, xy = 4*9 = 36.
x=9, y=4, xy = 9*4 = 36.

Since x=4 and x=9 each satisfy statement 1, insufficient.

Statement 2: x<5.
x could be any integer between 0 and 5.
Insufficient.

Statements 1 and 2 together:
Since x=1 does not satisfy statement 1, and x=4 is the only other perfect square less than 5, we know that x=4.
Sufficient.

The correct answer is C.

@Mitch
It seems that 1 is a perfect square:
https://www.tutorvista.com/math/is-1-a-perfect-square

https://www.math.com/school/subject1/les ... 1L9DP.html

https://www.mathsisfun.com/square-root.html

What did I miss..?

[GMATGuruNY]

If x and y are positive integers, what is the value of x ?
(1) x, y, and xy are distinct perfect squares less than 100.
(2) x<5

OA after discussion.

Statement 1: x, y and xy are distinct perfect squares less than 100.
x=1 is not possible because then xy = y, which does not satisfy the condition that y and xy be different values.
y=1 is not possible because then xy = x, which does not satisfy the condition that x and xy be different values.
The following combinations work:
x=4, y=9, xy = 4*9 = 36.
x=9, y=4, xy = 9*4 = 36.

Since x=4 and x=9 each satisfy statement 1, insufficient.

Statement 2: x<5.
x could be any integer between 0 and 5.
Insufficient.

Statements 1 and 2 together:
Since x=1 does not satisfy statement 1, and x=4 is the only other perfect square less than 5, we know that x=4.
Sufficient.

The correct answer is C.

@Mitch
It seems that 1 is a perfect square:
https://www.tutorvista.com/math/is-1-a-perfect-square

https://www.math.com/school/subject1/les ... 1L9DP.html

https://www.mathsisfun.com/square-root.html

What did I miss..?

The smallest perfect square is 1.
But statement 1 states that x, y and xy are distinct (meaning different) perfect squares. In order for xy to be distinct from x and y, neither x=1 nor y=1 can be used. To illustrate:

If x=1 and y=4, then xy = 1*4 = 4.
Since y and xy are the same value, the combination above does not satisfy the condition that x, y and xy be distinct perfect squares.

If y=1 and x=4, then xy = 4*1 = 4.
Since x and xy are the same value, the combination above does not satisfy the condition that x, y and xy be distinct perfect squares.

Hope the explanation above clarifies why x=1 cannot be used in statement 1.

[gig92]

If x and y are positive integers, what is the value of x ?
(1) x, y, and xy are distinct perfect squares less than 100.
(2) x<5

OA after discussion.

Statement 1: x, y and xy are distinct perfect squares less than 100.
x=1 is not possible because then xy = y, which does not satisfy the condition that y and xy be different values.
y=1 is not possible because then xy = x, which does not satisfy the condition that x and xy be different values.
The following combinations work:
x=4, y=9, xy = 4*9 = 36.
x=9, y=4, xy = 9*4 = 36.

Since x=4 and x=9 each satisfy statement 1, insufficient.

Statement 2: x<5.
x could be any integer between 0 and 5.
Insufficient.

Statements 1 and 2 together:
Since x=1 does not satisfy statement 1, and x=4 is the only other perfect square less than 5, we know that x=4.
Sufficient.

The correct answer is C.

@Mitch
It seems that 1 is a perfect square:
https://www.tutorvista.com/math/is-1-a-perfect-square

https://www.math.com/school/subject1/les ... 1L9DP.html

https://www.mathsisfun.com/square-root.html

What did I miss..?

The smallest perfect square is 1.
But statement 1 states that x, y and xy are distinct (meaning different) perfect squares. In order for xy to be distinct from x and y, neither x=1 nor y=1 can be used. To illustrate:

If x=1 and y=4, then xy = 1*4 = 4.
Since y and xy are the same value, the combination above does not satisfy the condition that x, y and xy be distinct perfect squares.

If y=1 and x=4, then xy = 4*1 = 4.
Since x and xy are the same value, the combination above does not satisfy the condition that x, y and xy be distinct perfect squares.

Hope the explanation above clarifies why x=1 cannot be used in statement 1.

Thanks for your time and yes it's GREAT ... I missed that "distinct"...