Question

The dimensions of a window are 156 cm \(\times\) 216 cm. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

Hint:

Whenever a problem asks for "maximum size" or "largest possible" pieces to be cut or fitted without remainder, it is a hint to calculate the H.C.F.

Answer 1

Step 1: Understanding the Concept:
To create complete squares of "maximum size" that fit exactly into the rectangular dimensions, the side of the square must be the Highest Common Factor (H.C.F.) of the length and width of the window.
Step 2: Detailed Explanation:
We need to find H.C.F.(156, 216).
Prime factorization of 156:
\[ 156 = 2 \times 78 = 2^{2} \times 39 = 2^{2} \times 3 \times 13 \]
Prime factorization of 216:
\[ 216 = 2 \times 108 = 2^{2} \times 54 = 2^{3} \times 27 = 2^{3} \times 3^{3} \]
H.C.F. is the product of the lowest powers of common prime factors:
\[ \text{H.C.F.} = 2^{2} \times 3 = 4 \times 3 = 12 \]
Thus, the side length of the maximum square is 12 cm.
Finding number of squares:
\[ \text{Number of squares} = \frac{\text{Area of window}}{\text{Area of one square}} \]
\[ \text{Number of squares} = \frac{156 \times 216}{12 \times 12} \]
\[ \text{Number of squares} = \left(\frac{156}{12}\right) \times \left(\frac{216}{12}\right) = 13 \times 18 = 234 \]
Step 3: Final Answer:
The maximum side length of the square is 12 cm and the total number of squares formed is 234.