$$\frac{x\cdot y}{x+y}=without\ remainder?$$
The question is asking if the sum of x and y can divide the product of x and y.
Statement 1=> x=y
$$\sin ce\ x=y,\ \frac{x\cdot y}{x+y}=\frac{x\cdot x}{x+x}=\frac{x^2}{2x}=\frac{x}{2}$$
The denominator is an even number; if x = even, then it is divisible without remainder but if x = odd then it is not divisible without remainder. Information given is not enough to provide answers to the question. Hence, statement 1 is NOT SUFFICIENT
Statement 2 => x = 2
$$\sin ce\ x=2;\ \frac{x\cdot y}{x+y}=\frac{2\cdot y}{2+y}=\frac{2y}{2+y}$$
The value of y is unknown and we cannot determine if the product of x and y is divisible by the sum of x and y. Hence, statement 2 is NOT SUFFICIENT
Combining both statements together =>
Statement 1 => x/2
Statement 2 => x=2
Since x=2, therefore, x/2 = 2/2 = 1
Thus, the product of two positive integers x and y is divisible by the sum of x and y. Both statements together are SUFFICIENT
Answer = option C
