CAT 2020 Slot 2 QA Question 9: The number of pairs of integers $(x, y)$ satisfying $x \geq y \geq -20$ and $2x + 5y = 99$ is

Question

The number of pairs of integers $(x, y)$ satisfying $x \geq y \geq -20$ and $2x + 5y = 99$ is

Accepted Answer

From the equation,

$x = \dfrac{99 - 5y}{2}$

Since $x \geq y$,
$\dfrac{99 - 5y}{2} \geq y$
$99 \geq 7y$
$y \leq 14$

Also $y \geq -20$. So $y$ can range from $-20$ to $14$.
For $x$ to be an integer, $99 - 5y$ must be even. Since $99$ is odd and $5y$ has the parity of $y$, $y$ must be odd.

The odd

The number of pairs of integers (x,y)(x, y)(x,y) satisfying x≥y≥−20x \geq y \geq -20x≥y≥−20 and 2x+5y=992x + 5y = 992x+5y=99 is

The number of pairs of integers $(x, y)$ satisfying $x \geq y \geq -20$ and $2x + 5y = 99$ is