Sure! Let's solve the problem step-by-step.
### Step 1: Convert the dimensions to a consistent unit
First, we need to convert the dimensions of the room from meters and centimeters to centimeters only for easier calculations.
- Length: [tex]\(7 \, \text{meters} + 83 \, \text{centimeters}\)[/tex]
[tex]\[
1 \, \text{meter} = 100 \, \text{centimeters}
\][/tex]
Therefore,
[tex]\[
7 \, \text{meters} \times 100 = 700 \, \text{centimeters}
\][/tex]
So, the length in centimeters is:
[tex]\[
700 \, \text{cm} + 83 \, \text{cm} = 783 \, \text{cm}
\][/tex]
- Breadth: [tex]\(5 \, \text{meters} + 13 \, \text{centimeters}\)[/tex]
[tex]\[
1 \, \text{meter} = 100 \, \text{centimeters}
\][/tex]
Therefore,
[tex]\[
5 \, \text{meters} \times 100 = 500 \, \text{centimeters}
\][/tex]
So, the breadth in centimeters is:
[tex]\[
500 \, \text{cm} + 13 \, \text{cm} = 513 \, \text{cm}
\][/tex]
### Step 2: Find the Greatest Common Divisor (GCD)
We need to find the greatest common divisor (GCD) of these two dimensions because the GCD will give us the largest possible side length of the square tile that can cover the floor completely without overlaps or gaps.
Given:
- Length = [tex]\(783 \, \text{cm}\)[/tex]
- Breadth = [tex]\(513 \, \text{cm}\)[/tex]
The GCD of 783 and 513 is 27.
### Step 3: Conclusion
The dimension of the largest square tile that can be fixed on the floor completely is:
[tex]\[
\boxed{27 \, \text{cm}}
\][/tex]
