The longest tape that can measure the dimensions of a room exactly is found using the concept of the greatest common divisor (GCD). The dimensions of the room are given as:
- Length: 825 cm
- Breadth: 675 cm
- Height: 450 cm
To find the GCD of these three numbers, we can break each dimension down into its prime factors:
Factorization:
- 825 = 3 x 5^2 x 11
- 675 = 3^3 x 5^2
- 450 = 2 x 3^2 x 5^2
Now, we need to identify the common prime factors:
– The prime factor 3 appears in the factorization of all three dimensions, and the minimum power is 1 (from 825).
– The prime factor 5 also appears, and the minimum power is 2 (from all three).
– The prime factor 2 does not appear in 825 or 675.
So, the GCD can be calculated as:
GCD = 3^1 x 5^2 = 3 x 25 = 75 cm
This means that the longest tape that can measure all three dimensions of the room exactly is 75 cm.