If x and y are positive integers does x equal to y?

Question

(1) The sum of the greatest common divisor of x and y and the least common multiple of x and y equals to the sum of x and y
(2) The greatest common divisor of x and y equals to the least common multiple of x and y

M36-83

Bunuel · Accepted Answer

Official Solution:

If  x  and  y  are positive integers does  x  equal to  y ? 
 
(1) The sum of the greatest common divisor of  x  and  y  and the least common multiple of  x  and  y  equals to the sum of  x  and  y 
 
The above is obviously true when  x = y . For example, if  x=y=3 , then the  GCD(3, 3)=3  and the  LCM(3, 3)=3 , so  GCD(3, 3)+LCM(3, 3)=3+3 .
 
Let's see whether the above could be true when  x 
eq y . Say  x=1  and  y=2 , then  GCD(1, 2)=1  and the  LCM(1, 2)=2 , so  GCD(1, 2)+LCM(1, 2)=1+2 . So,  x 
eq y  case is also possible for this statement.
 
Not sufficient.
 
(2) The greatest common divisor of  x  and  y  equals to the least common multiple of  x  and  y 
 
If  x 
eq y , then obviously  GCD(x, y) < LCM(x, y) . For example, say  x < y , then the  GCD(x, y) \leq x  and the  LCM(x, y) \geq y , so  GCD(x, y) < LCM(x, y) . Thus,  x=y . Sufficient.

Answer: B

XSatishX · Answer

x and y are two positive integers. Is x equals to y?
Statement 1:  Not Sufficient 
GCD+LCM = x+y
If x=8 and y =4, than LCM = 8 and GCD =4.Hence LCM+GCD=x+y but x is not equal to y. So we get a No here.
Now if x=3 and y=3 then LCM=3 and GCD=3 and hence LCM+GCD=x+y and x=y. So we get a Yes here.
Hence this statement is not sufficient.

Statement 2:  Sufficient 
GCD = LCM
Let GCD=LCM=a.
We know LCM*GCD = x*y
Hence x*y = a*a
Also, x and y are positive integers. So, x and y will be equal to a.

Answer  B

RajatJ79 · Answer

If x and y are positive integers does x equal to y?

Solution  -

Considering Statement (1) alone -
 (1) The sum of the greatest common divisor of x and y and the least common multiple of x and y equals to the sum of x and y 
If x = 4 and y = 2, GCD = 2 and LCM = 4. In this case, sum of the GCD and LCM of x and y is equal to sum of x and y. Answer to question will be Yes.
If x = 4 and y = 4, GCD = 2 and LCM = 4. In this case, sum of the GCD and LCM of x and y is NOT equal to sum of x and y. Answer to question will be No.

Statement (1) does not have a consistent answer. Statement (1) alone is not sufficient.

Considering Statement (2) alone -
 (2) The greatest common divisor of x and y equals to the least common multiple of x and y. 
If x = 2 and y = 2, GCD = 2 and LCM = 2. Similar will be the case with prime numbers such as 3, 5, 7, etc.
 GCD of any positive integers x and y will be equal to LCM of x and y only when both x and y are the same prime numbers. i.e when both x and y are equal and prime.

Statement (2) alone is sufficient. It gives a Yes answer to the question.

Answer should be Option B.

stne · Answer

Plugging in method.

(1)If  x=2   and  y=2  yes 
 if  x=1  and  y =2   no 
 INSUFF.

(2) Only possible when both  x  and  y  are equal.
i.e.  x=2\hspace{2mm} y = 2 
  x=3 \hspace{2mm} y =3

SUFF.

Ans- B

Hope it's clear.

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 does x equal to y?

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 Data Sufficiency (DS) Questions

If x and y are positive integers does x equal to y?

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