Step 1: Determine Total Tank Capacity
The rates of flow (efficiency) are given as $A : B : C = 4 : 9 : 36$.
Let the rates be $4k, 9k,$ and $36k$ units per hour.
Since Pipe $A$ fills the tank in 15 hours, the total capacity of the tank is:
$\text{Capacity} = \text{Rate of } A \times \text{Time} = 4k \times 15 = 60k$ units.

Step 2: Combined Rate of All Pipes
When all three pipes work together, their combined rate of flow is:
$R_{total} = 4k + 9k + 36k = 49k$ units/hour.

Step 3: Calculate Time in Minutes
$\text{Time taken (hours)} = \frac{\text{Capacity}}{R_{total}} = \frac{60k}{49k} = \frac{60}{49}$ hours.
To convert this to minutes, multiply by 60:
$\text{Time in minutes} = \frac{60}{49} \times 60 = \frac{3600}{49}$ minutes.
$\frac{3600}{49} \approx 73.469$ minutes.

The nearest integer value is $\mathbf{73}$.