If x and y are positive integers less than 10

Question

If x and y are positive integers less than 10, which of the following COULD be true?

i) √x + √y = √(x + y)
ii) y√x = x√y
iii) √x - √y = √(x - y)

A) i only
B) i and ii only
C) i and iii only
D) ii and iii only
E) i,ii and iii

SSM700plus · Accepted Answer

Solution - 
Well I am amazed to see the answer as E. I tried all the approach, not sure where I am missing the trick.

It is COULD Be question. So we just need one pair of value - (x,y) which satisfies the condition.

Here x, y are +ve Integer and less than 10.

i) √x + √y = √(x + y).
Tried different values of x & y but couldn't get one such value for which the equation is true. So I tried algebric method.
√x + √y = √(x + y).
Squaring on both sides.
 (√x + √y )^2= (x + y). 
x + y +2√x√y = x + y.
We are left with 2√x√y = 0.
This is possible only when x or y is zero. but x & y are +ve integer. Hence not true for any value.

ii) y√x = x√y.
Let's take x=1 and y=1 . 
So y√x = 1*1 = 1. 
And x√y = 1*1 = 1.
Hence it could be true.

iii) √x - √y = √(x - y)
Let's take x=1 and y=1 . 
So √x - √y = √1 - √1 = 0. 
And √(x - y) = √(1 - 1)= 0.
Hence it could be true.

Hence I will go with D.

Please let me know where I am missing the trick.
Thanks.

BrentGMATPrepNow · Answer

My bad! The correct answer is, indeed, D. I've edited it. 
Kudos to you!!

Cheers,
Brent

AkshdeepS · Answer

Best way to deal with such problems is try some values and POE:

Try x and y = 1, we get  \sqrt{1} +\sqrt{1} = \sqrt{1 + 1}

2 = 1.4 --- Not true.

Try x = 2, y = 3, we get  \sqrt{2} + \sqrt{3} = \sqrt{2 + 3}

3.1 = 2.2 ---- Not true.

This way you can check for few more values and you will realize (I) is not true.

Therefore, you can eliminate all options that have (I).

A,B,C, and E.

Hence D.

BrentGMATPrepNow · Answer

When I  scan  the statements, I see that ii and iii look easier, so I'll start with those.

ii) y√x = x√y 
We can quickly see that, if x = y, then this statement is TRUE. 
For example, if x = 1 and y = 1, we get 1√1 = 1√1, which is true. 
So,  statement ii COULD be true . 
Check the answer choices......ELIMINATE A and C

iii) √x - √y = √(x - y) 
Once again, this statement is TRUE when x = y. 
For example, if x = 1 and y = 1, we get √1 - √1 = √(1 - 1), which is true. 
So,  statement iii COULD be true . 
Check the answer choices......ELIMINATE B

i) √x + √y = √(x + y) 
This is a tough one. 
I can't think of any values for x and y that make this statement true, so now what? 
Well, we might just conclude that statement i cannot be true. 
However, we can do a bit of algebra to further convince ourselves.

Given: √x + √y = √(x + y)
Square both sides to get: (√x + √y)² =  ²
Expand to get: x + 2√(xy) + y = x + y
Subtract x and y from both sides to get: 2√(xy) = 0
This means that xy = 0, but  xy cannot equal 0  since we're told that x and y are POSITIVE integers. 
So,  statement i cannot be true.

Answer: D

Cheers,
Brent

ScottTargetTestPrep · Answer

Solution:

If x = y, we see that Roman numerals II and III are true. However, Roman numeral I can never be true when x and y are positive integers. That is because, when we square both sides of the equation, we have:

LHS = (√x + √y)^2 = x + 2√x√y + y and RHS = (√(x + y))^2 = x + y

We see that, when each side is squared, the LHS is 2√x√y more than the RHS. They still can be equal if either x or y is 0. However, we are given that x and y are positive; therefore, 2√x√y can never be 0. In other words, (√x + √y)^2 > (√(x + y))^2 or √x + √y > √(x + y). So √x + √y = √(x + y) is a false statement when x and y are positive.

Answer: D

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If x and y are positive integers less than 10

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If x and y are positive integers less than 10

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards