Q25 / 55 · KVPY 2020 (SA) · Mathematics

The number of ordered pairs $(x, y)$ of positive integers satisfying $2^{x} + 3^{y} = 5^{x y}$ is

Solution

$2^{x} + 3^{y} = 5^{x y}$

Clearly $x = y = 1$ satisfy the relation

Take $x > y$

$3^{x} > 3^{y}$

$2^{x} + 3^{x} > 2^{x} + 3^{y}$

$2^{x} + 3^{x} > 5^{x y}$

this is false

$5^{x y} > 5^{x} = {(2 + 3)}^{x} > 2^{x} + 3^{x}$

Similarly there is no solution for $x < y$

Hence $x = y$ Hence only $x = y = 1$ satisfy the given equation ( $2^{x} + 3^{x} = 5^{x y}$ is not true for $x \in N - {1}$ )

Hence, the correct answer is option: 'A'.