The length, breadth and height of a room are $8$ m $50$ cm, $6$ m $25$ cm and $4$ m $75$ cm respectively. Find the length of the longest rod that can measure the dimensions of the room exactly.

The given dimensions are $825$cm, $675$cm & $450$cm.

Now to find the length of room we need a common factor of $825,675,450$

Using Prime factorization:

$825=3\times 5^2\times 11$

$675=3^2\times 5^2$

$450=2\times 3^2\times 5^2$

Now from above

HCF$=3\times 5^2=75$

$\therefore$ The length of longest rod to measure the length of room is $75$cm.