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
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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.
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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.
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