Let M be the maximum value of the product of two positive integers...

Given,

Sum of two integer is $66$, so one number will be $x$ and other will be $66 - x$,

And given $M$ is maximum value of their product,

So let $y = x\left(66 - x\right)$

$⇒ y = 66x - {x}^{2}$

Now differentiating to find maxima and minima we get,

$\dfrac{dy}{dx} = 66 - 2x$

Now equating with zero to find point of maxima as $y = 66x - {x}^{2}$ represents a downward parabola so it will give maxima,

So $\dfrac{dy}{dx} = 0 ⇒ 66 - 2x = 0 ⇒ x = 33$,

Hence, the value of $M = 33 \times 33 = 1089$

Now solving $x\left(66 - x\right) \ge \dfrac{5M}{9}$

$⇒ x\left(66 - x\right) \ge \dfrac{5 \times 1089}{9}$

$⇒ x\left(66 - x\right) \ge 605$

$⇒ {x}^{2} - 66x + 605 \le 0$

$⇒ \left(x - 11\right)\left(x - 55\right) \le 0$

So $x \in \left[11, 55\right] \to$total $45$ numbers,

Now for probability of $A$, favourable outcomes will be $x = 3k ⇒ x = \left\{12, 15, 18, . . . . . 54\right\} \to$ total $15$ numbers,

So probability will be $\dfrac{15}{45} = \dfrac{1}{3}$