Question

If $a>0$ and $b>0$, a new operation $\\abla$ is defined as follows:$$a \\abla b = \dfrac{a + b}{1 + ab}.$$For example,$$3 \\abla 6 = \dfrac{3 + 6}{1 + 3 \times 6} = \dfrac{9}{19}.$$For some values of $x$ and $y$, the value of $x \\abla y$ is equal to $\dfrac{x + y}{17}$. How many possible ordered pairs of positive integers $x$ and $y$ are there for which this is true?

Answer 1

This happens when

1 + a b = 17 ==> a b = 16

With a and b both positive integers, and 16 = 2^4, we can have

• a = 1 and b = 16

• a = 2 and b = 8

• a = b = 4

and vice versa. So there are 5 possible ordered pairs.