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

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.

Bunuel · 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

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

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