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

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 \(x \neq y\). 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
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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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