HOT Competition 1 Sep/8PM: If x and y are positive integers, is x + y

Official Solution:


If \(x\) and \(y\) are positive integers, is \(x + y\) an even integer?

(1) The product of the greatest common factor of \(x\) and \(y\) and the least common multiple of \(x\) and \(y\) is even

The product of the GCF and LCM of two positive integers is equal to the product of those integers. So, we are given that \(GCF(x,y)*LCM(x,y)=xy=even\). Now, \(xy=even\) means that at least one of the unknowns is even. If both are even, then \(x + y=even\) but if only one of them is even and another is odd, then \(x + y=odd\). For example, consider \(x=2\) and \(y=4\) for an YES answer and \(x=2\) and \(y=3\) for a NO answer. Not sufficient.

(2) The greatest common factor of \(x\) and \(y\) is 1

Clearly insufficient. For example, consider \(x=3\) and \(y=1\) for an YES answer and \(x=3\) and \(y=2\) for a NO answer. Not sufficient.

(1)+(2) From (1) we got that either both unknowns are even or one is even and another is odd. The first case is ruled out by (2), since if both are even, then they both would share 2 as a common factor, while (2) says that the greatest common factor of \(x\) and \(y\) is 1. Thus, one of the unknowns is even and another is odd: \(x + y=odd\). Sufficient.


Answer: C