Problem:
The length, breadth, and height of a room are 8 m 25 cm, 6 m 75 cm, and 4 m 50 cm respectively. Determine the longest tape, which can measure the three dimensions of the room exactly.

Solution:
To find the longest tape that can measure the three dimensions of the room exactly, we need to consider the greatest common divisor (GCD) of the lengths.

Step 1: Convert the measurements to the same unit
First, we need to convert all the measurements to the same unit. Let's convert all the measurements to centimeters (cm) for simplicity.

Length = 8 m 25 cm = (8 x 100) cm + 25 cm = 825 cm
Breadth = 6 m 75 cm = (6 x 100) cm + 75 cm = 675 cm
Height = 4 m 50 cm = (4 x 100) cm + 50 cm = 450 cm

Step 2: Find the GCD of the measurements
Now, we need to find the GCD of the three measurements: 825 cm, 675 cm, and 450 cm.

To find the GCD, we can use the Euclidean algorithm:
- Divide the largest number (825 cm) by the smallest number (450 cm) and find the remainder.
- Then, divide the divisor (450 cm) by the remainder and find the new remainder.
- Continue this process until the remainder is zero.

Using this algorithm, we find:
825 cm ÷ 450 cm = 1 remainder 375 cm
450 cm ÷ 375 cm = 1 remainder 75 cm
375 cm ÷ 75 cm = 5 remainder 0 cm

The GCD of 825 cm, 675 cm, and 450 cm is 75 cm.

Step 3: Determine the longest tape
To determine the longest tape that can measure the three dimensions of the room exactly, we need to find the LCM (Least Common Multiple) of the measurements.

The LCM is the smallest multiple that is divisible by all the given numbers.

To find the LCM, we can use the formula:
LCM = (Product of the numbers) ÷ GCD

The product of the measurements is:
825 cm * 675 cm * 450 cm = 255,937,500 cm^3

Therefore, the longest tape that can measure the three dimensions of the room exactly is:
LCM = 255,937,500 cm^3 ÷ 75 cm = 3,419,166.67 cm

So, the longest tape needed is approximately 3,419,166.67 cm.