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