Solution:

Given dimensions of the room are:

Length = 6m 75cm = 675cm
Breadth = 4m 50cm = 450cm
Height = 2m 25cm = 225cm

To find the length of the longest measuring rod which can measure the dimensions in exact number of times, we need to find the HCF (Highest Common Factor) of the given dimensions.

Step 1: Prime factorize the given dimensions

Length = 675 = 3 x 3 x 3 x 5 x 5
Breadth = 450 = 2 x 3 x 3 x 5 x 5
Height = 225 = 3 x 3 x 5 x 5

Step 2: Identify the common factors

The common factors are 3, 5, and 5.

Step 3: Multiply the common factors

3 x 5 x 5 = 75

Therefore, the length of the longest measuring rod which can measure the dimensions in exact number of times is 75cm.

Explanation:

The given dimensions of the room are expressed in centimeters. To find the length of the longest measuring rod which can measure the dimensions in exact number of times, we need to find the HCF (Highest Common Factor) of the given dimensions.

The HCF is the highest number that divides all the given numbers without leaving any remainder. In other words, it is the largest number that is a factor of all the given numbers.

To find the HCF, we need to prime factorize the given dimensions. Prime factorization is the process of expressing a number as a product of prime factors.

Once we have prime factorized the given dimensions, we need to identify the common factors. The common factors are the factors that are common to all the given numbers.

Finally, we need to multiply the common factors to get the HCF. The HCF is the length of the longest measuring rod which can measure the dimensions in exact number of times.

In this case, the HCF is 75cm. So, a measuring rod of length 75cm can measure the dimensions of the room in exact number of times.