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 Author: Bunuel [ 28 Jan 2021, 13:46 ] Post subject: If x and y are positive integers does x equal to y? 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(2) The greatest common divisor of x and y equals to the least common multiple of x and yM36-83

 Author: Bunuel [ 11 Nov 2021, 02:39 ] Post subject: Re: If x and y are positive integers does x equal to y? Bunuel wrote: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(2) The greatest common divisor of x and y equals to the least common multiple of x and yM36-83Official 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 $$x \neq y$$. 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, $$x \neq y$$ 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 $$x \neq y$$, 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

 Author: XSatishX [ 28 Jan 2021, 19:25 ] Post subject: Re: If x and y are positive integers does x equal to y? x and y are two positive integers. Is x equals to y?Statement 1: Not SufficientGCD+LCM = x+yIf 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: SufficientGCD = LCMLet GCD=LCM=a.We know LCM*GCD = x*yHence x*y = a*aAlso, x and y are positive integers. So, x and y will be equal to a.Answer B

 Author: RajatJ79 [ 30 Jan 2021, 00:39 ] Post subject: Re: If x and y are positive integers does x equal to y? 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 yIf 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.

 Author: stne [ 06 Feb 2021, 08:49 ] Post subject: Re: If x and y are positive integers does x equal to y? Bunuel wrote: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(2) The greatest common divisor of x and y equals to the least common multiple of x and yPlugging 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- BHope it's clear.

 Author: bumpbot [ 30 Dec 2022, 17:00 ] Post subject: Re: If x and y are positive integers does x equal to y? Hello from the GMAT Club BumpBot!Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

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