## Sum of Two Positive Integers

$N$ and $Y$ are positive integers, $N\neq Y$. If $N\times Y\div 2=N+Y$, what is the sum $N+Y$?
Source: NCTM Mathematics Teacher 2008

SOLUTION
$N\times Y\div 2=N+Y$
$NY=2(N+Y)$
$NY-2Y=2N$
$Y(N-2)=2N$
$Y=\dfrac{2N}{N-2}$
Table of $N$ and $Y$ values
$N\qquad Y$
$1\qquad -2$
$2\qquad\text{undefined}$
$3\qquad 6$
$4\qquad 4$
$5\qquad 3.33$
$6\qquad 3$
$7\qquad 2.8$
$8\qquad 2.67$
$9\qquad 2.57$
$10\qquad 2.5$
$\cdots$
The graph of $Y$ as a function of $N$ shows a horizontal asymptote at $Y=2$. For $N>6,\: 2.

Two possible values of $(N,Y):(3,6)$ or $(6,3)$.
$N+Y=3+6=9$

Answer: $9$